mathematical studies up to innperseqs diagram

6 files changed, 533 insertions, 724 deletions

diff --git a/aux.otx b/aux.otx
index ef7b8db..e93e1f5 100644
--- a/aux.otx
+++ b/aux.otx
@@ -100,6 +100,12 @@
 \def\unstep{\advance\leftskip by -\parindent}
 
 % ---
+\long\def\sysrules#1{{
+ \parindent=0pt\parskip=0.5em
+ \hangindent=1em\hangafter=1
+ #1\par\vskip 0.5em}}
+\def\hi{\hangindent=1em}
+% ---
 
 \fontfam[Pagella]
 \typosize[10/13]
diff --git a/blueprint.otx b/blueprint.otx
index 76796db..9f1266b 100644
--- a/blueprint.otx
+++ b/blueprint.otx
@@ -107,9 +107,9 @@ colophon goes here
 
 \part Para-Science
 \input essays/dissociation_physics.otx
-% \input{essays/mathematical_studies.tex}
-% \input{essays/post_formalism_memories.tex}
-% \input{essays/studies_in_constructed_memories.tex}
+\input essays/mathematical_studies.otx
+%\input{essays/post_formalism_memories.tex}
+%\input{essays/studies_in_constructed_memories.tex}
 
 %\part{The New Modality}
 %\input{essays/energy_cube1966.tex}
diff --git a/essays/mathematical_studies.tex b/essays/mathematical_studies.otx
index ffb914b..e386199 100644
--- a/essays/mathematical_studies.tex
+++ b/essays/mathematical_studies.otx
@@ -1,13 +1,15 @@
-\chapter{1966 Mathematical Studies: Introduction}
+\chapter 1966 Mathematical Studies
+
+% \fancyhead{} \fancyfoot{} \fancyfoot[LE,RO]{\thepage}
+% \fancyhead[LE]{\textsc{Mathematical Studies (1966)}} \fancyhead[RO]{\textit{Introduction}}
+\sec Introduction
 
-\fancyhead{} \fancyfoot{} \fancyfoot[LE,RO]{\thepage}
-\fancyhead[LE]{\textsc{Mathematical Studies (1966)}} \fancyhead[RO]{\textit{Introduction}}
 
 Pure mathematics is the one activity which is intrinsically formalistic. It is the one activity which brings out the practical value of formal manipulations. Abstract games fit in perfectly with the tradition and rationale of pure mathematics; whereas they would not be appropriate in any other discipline. Pure mathematics is the one activity which can appropriately develop through innovations of a formalistic character. 
 
 Precisely because pure mathematics does not have to be immediately practical, there is no intrinsic reason why it should adhere to the normal concept of logical truth. No harm is done if the mathematician chooses to play a game which is indeterminate by normal logical standards. All that matters is that the mathematician clearly specify the rules of his game, and that he not make claims for his results which are inconsistent with his rules. 
 
-Actually, my pure philosophical writings discredit the concept of logical truth by showing that there are flaws inherent in all non-trivial language. Thus, no mathematics has the logical validity which was once claimed for mathematics. From the ultimate philosophical standpoint, all mathematics is as \enquote{indeterminate} as the mathematics in this monograph. All the more reason, then, not to limit mathematics to the normal concept of logical truth. 
+Actually, my pure philosophical writings discredit the concept of logical truth by showing that there are flaws inherent in all non-trivial language. Thus, no mathematics has the logical validity which was once claimed for mathematics. From the ultimate philosophical standpoint, all mathematics is as \dq{indeterminate} as the mathematics in this monograph. All the more reason, then, not to limit mathematics to the normal concept of logical truth. 
 
 Once it is realized that mathematics is intrinsically formalistic, and need not adhere to the normal concept of logical truth, why hold back from exploring the possibilities which are available? There is every reason to search out the possibilities and present them. Such is the purpose of this monograph. 
 
diff --git a/essays/post_formalism_memories.otx b/essays/post_formalism_memories.otx
new file mode 100644
index 0000000..6219065
--- /dev/null
+++ b/essays/post_formalism_memories.otx
@@ -0,0 +1,518 @@
+%\newcommand{\midheading}[1]{
+%	{ \vskip 1em \centering \large \textsc{#1} \par \vskip 1em }}
+
+
+\sec Post-Formalism in Constructed Memories
+ 
+\section{Post-Formalist Mathematics}
+
+%\fancyhead{} \fancyfoot{} \fancyfoot[LE,RO]{\thepage}
+%\fancyhead[LE]{\textsc{Post-Formalism in Constructed Memories}} \fancyhead[RO]{\textit{Post-Formalist Mathematics}}
+
+Over the last hundred years, a philosophy of pure mathematics has grown up which I prefer to call \dq{formalism.} As Willard Quine says in the fourth section of his essay \essaytitle{Carnap and Logical Truth,} formalism was inspired by a series of developments which began with non-Euclidian geometry. Quine himself is opposed to formalism, but the formalists have found encouragement in Quine's own book, \booktitle{Mathematical Logic}. The best presentation of the formalist position can be found in Rudolph Carnap's \booktitle{The Logical Syntax of Language}. As a motivation to the reader, and as a heuristic aid, I will relate my study to these two standard books. (It will help if the reader is thoroughly familiar with them.) it is not important whether Carnap, or Quine, or formalism---or my interpretation of them---is \dq{correct,} for this essay is neither history nor philosophy. I am using history as a bridge, to give the reader access to some extreme mathematical innovations. 
+
+The formalist position goes as follows. Pure mathematics is the manipulation of the meaningless and arbitrary, but typographically well-defined ink-shapes on paper \sq{$w$,} \sq{$x$,} \sq{$y$,} \sq{$z$,} \sq{{\tt '},} \sq{$($,} \sq{$)$,} \sq{$\downarrow$,} and \sq{$\in$.} These shapes are manipulated according to arbitrary but well-defined mechanical rules. Actually, the rules mimic the structure of primitive systems such as Euclid's geometry. There are formation rules, mechanical definitions of which concatenations of shapes are \dq{\term{sentences}.} One sentence is \sq{$((x) (x\in x) \downarrow (x) (x\in x))$.} There are transformation rules, rules for the mechanical derivation of sentences from other sentences. The best known transformation rule is the rule that $\psi$ may be concluded from $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$; where \sq{$\supset$} is the truth-functional conditional. For later convenience, I will say that $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$ are \dq{\term{impliors},} and that $\psi$ is the \dq{\term{implicand}.} Some sentences are designated as \dq{\term{axioms}.} A \dq{\term{proof}} is a series of sentences such that each is an axiom or an implicand of preceding sentences.  The last sentence in a proof is a \dq{\term{theorem}.} 
+
+This account is ultra-simplified and non-rigorous, but it is adequate for my purposes. (The reader may have noticed a terminological issue here. For Quine, an implication is merely a logically true conditional. The rules which are used to go from some statements to others, and to assemble proofs, are rules of inference. The relevant rule of inference is the \term{modus ponens};\ednote{i.e., \dq{$P$ implies $Q$. $P$ is true. Therefore, $Q$ must also be true.}} $\psi$ is the ponential of $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$. What I am doing is to use a terminology of implication to talk about rules of inference and ponentials. The reason is that the use of Quine's terminology would result in extremely awkward formulations. What I will be doing is sufficiently transparent that it can be translated into Quine's terminology if necessary. My results will be unaffected.) The decisive feature of the arbitrary game called \dq{mathematics} is as follows. A sentence-series can be mechanically checked to determine whether it is a proof. But there is no mechanical method for deciding whether a sentence is a theorem. Theorems, or rather their proofs, have to be puzzled out, to be discovered. in this feature lies the dynamism, the excitement of traditional mathematics. Traditional mathematical ability is the ability to make inferential discoveries. 
+
+A variety of branches of mathematics can be specialized out from the basic system. Depending on the choices of axioms, systems can be constructed which are internally consistent, but conflict with each other. A system can be \dq{interpreted,} or given a meaning within the language of a science such as physics. So interpreted, it may have scientific value, or it may not. But as pure mathematics, all the systems have the same arbitrary status. 
+
+By \dq{formalist mathematics} I will mean the present mathematical systems which are presented along the above lines. Actually, as many authors have observed, the success of the non-Euclidian \dq{imaginary} geometries made recognition of the game-like character of mathematics inevitable.  Formalism is potentially the greatest break with tradition in the history of mathematics. In the \essaytitle{Foreward} to \booktitle{The Logical Syntax of Language}, Carnap brilliantly points out that mathematical innovation is still hindered by the widespread opinion that deviations from mathematical tradition must be justified---that is, proved to be \dq{correct} and to be a faithful rendering of \dq{the true logic.} According to Carnap, we are free to choose the rules of a mathematical system arbitrarily. The striving after correctness must cease, so that mathematics will no longer be hindered. \dq{\e{Before us lies the boundless ocean of unlimited possibilities.}} In other words, Carnap, the most reputable of academicians, says you can do anything in mathematics. Do not worry whether whether your arbitrary game corresponds to truth, tradition, or reality: it is still legitimate mathematics. Despite this wonderful \uline{Principle of Tolerance} in mathematics, Carnap never ventured beyond the old ink-on-paper, axiomatic-deductive structures. I, however, have taken Carnap at his word. The result is my \dq{post-formalist mathematics.} I want to stress that my innovations have been legitimized in advance by one of the most reputable academic figures of the twentieth century. 
+
+Early in 1961, I constructed some systems which went beyond formalist mathematics in two respects. 
+\begitems\style n
+* My sentential elements are physically different from the little ink-shapes on paper used in all formalist systems. My sentences are physically different from concatenations of ink-shapes. My transformation rules have nothing to do with operations on ink-shapes. 
+* My systems do not necessarily follow the axiomatic-deductive, sen\-ten\-ce-implication-axiom-proof-theorem structure. 
+\enditems
+\vskip 0.5em
+
+Both of these possibilities, by the way, are mentioned by Carnap in \essaytitle{Languages as Calculi.}\ednote{Also in \booktitle{The Logical Syntax of Language}.} A \dq{post-formalist system,} then, is a formalist system which differs physically from an ink-on-paper system, or which lacks the axiomatic-deductive structure. 
+
+As a basis for the analysis of post-formalist systems, a list of structural properties of formalist systems is desirable. Here is such a list. By \dq{implication} I will mean simple, direct implication, unless I say otherwise. 
+\vskip 0.5em
+\begitems\style i
+* A sentence can be repeated at will. 
+* The rule of implication refers to elements of sentences: sentences are structurally composite. 
+* A sentence can imply itself. 
+* The repeat of an implior can imply the repeat of an implicand: an implication can be repeated. 
+* Different impliors can imply different implicands. 
+* Given two or three sentences, it is possible to recognize mechanically whether one or two directly imply the third. 
+* No axiom is implied by other, different axioms. 
+* The definition of \dq{proof} is the standard definition, in terms of implication, given early in this essay. 
+* Given the axioms and some other sentence, it is not possible to recognize mechanically whether the sentence is a theorem.  Compound indirect implication is a puzzle. 
+\enditems
+\vskip 0.5em
+
+Now for the first post-formalist system. 
+
+\secc \sysname{Illusions}
+
+\sysrules{
+A \term{sentence} is the page (page \pgref[illusions], with figure \ref[illusions] on it) so long as the apparent, perceived ratio of the length of the vertical line to that of the horizontal line (the statement's \dq{associated ratio}) does not change. (Two sentences are the \dq{same} if end only if their associated ratios are the same.) 
+
+\hi A sentence $Y$ is \term{implied by} a sentence $X$ if and only if $Y$ is the same as $X$, or else $Y$ is, of all the sentences one ever sees, the sentence having the associated ratio next smaller than that of $X$. 
+
+\hi Take as the \term{axiom} the first sentence one sees. 
+
+\hi \e{Explanation}: The figure is an optical illusion such that the vertical line normally appears longer than the horizontal line, even though their lengths are equal. One can correct one's perception, come to see the vertical line as shorter relative to the horizontal line, decrease the associated ratio, by measuring the lines with a ruler to convince oneself that the vertical line is not longer than the other, and then trying to see the lines as equal in length; constructing similar figures with a variety of real (measured) ratios and practicing judging these ratios; and so forth.}
+
+\midinsert
+{\leftskip=0pt plus1fil\rightskip=0pt plus1fil
+\picw=4in\inspic{illusions.png} 
+	\cskip
+	\caption/f[illusions] The sentence for \sysname{Illusions}.
+	\par}
+\endinsert
+
+\sysname{Illusions} has Properties 1, 3--5, and 7--8. Purely to clarify this fact, the following sequence of integers is presented as a model of the order in which associated ratios might appear in reality. (The sequence is otherwise totally inadequate as a model of \sysname{Illusions.}) $4\medspace2\medspace1$; $4\medspace2$; $5\medspace4\medspace2\medspace1$; $4\medspace3\medspace1$. The implication structure would then be as shown in figure \ref[illusionstructure].
+
+\midinsert
+{\leftskip=0pt plus1fil\rightskip=0pt plus1fil
+\picw=4.5in\inspic{illusionstructure.png} \cskip
+	\caption/f[illusionstructure] Example implication structure for \sysname{Illusions}.
+\par}
+\endinsert
+
+The axiom would be 4, and 5 could not appear in a proof. \sysname{Illusions} has Property 1 on the basis that one can control the associated ratio. Turning to Property 4, it is normally the case that when an implication is repeated, a given occurrence of one of the sentences involved is unique to a specific occurrence of the implication. In \sysname{Illusions,} however, if two equal sentences are next smaller than $X$, the occurrence of $X$ does not uniquely belong to either of the two occurrences of the implication. Compare figure \ref{thestructure}, where the occurrence of `$t$' is not unique to either occurrence of `$the$'.  Subject to this explanation, \sysname{Illusions} has Property 4. \sysname{Illusions} has Property 8, but it goes without saying that the type of implication is not \term{modus ponens}. Properties 3, 5, and 7 need no comment. As for Property 2, the rule of implication refers to a property of sentences, rather than to elements of sentences. The interesting feature of \sysname{Illusions} is that it reverses the situation defined by Properties 6 and 9. Compound indirect implication is about the same as simple implication. The only difference is the difference between being smaller and being next smaller. And there is only one axiom (per person). 
+
+\midinsert
+{\leftskip=0pt plus1fil\rightskip=0pt plus1fil
+	\table{ccc}{t & h & e \cr h &   &   \cr e &   &}
+	\caption/f[thestructure] Structure with shared node.
+\par}
+\endinsert
+
+Simple direct implication, however, is subjective and illusive. It essentially involves changing one's perceptions of an illusion. The change of associated ratios is subjective, elusive, and certainly not numerically measurable. Then, the order in which one sees sentences won't always be their order in the implications and proofs. And even though one is exposed to all the sentences, one may have difficulty distinguishing and remembering them in consciousness. If I see the normal illusion, then manage to get myself to see the lines as being of equal length, I know I have seen a theorem. What is difficult is grasping the steps in between, the simple direct implications. If the brain contains a permanent impression of every sensation it has received, then the implications objectively exist; but they may not be thinkable without neurological techniques for getting at the impressions. In any case, \dq{proof} is well-defined in some sense---but proofs may not be thinkable. \sysname{Illusions} is, after all, not so much shakier in this respect than even simple arithmetic, which contains undecidable sentences and indefinable terms. 
+
+In \booktitle{The Logical Syntax of Language}, Carnap distinguishes pure syntax and descriptive syntax; and says that pure syntax should be independent of notation, and that every system should be isomorphic to some ink-on-paper system. In so doing, Carnap violates his own \uline{Principle of Tolerance}. Consider the following trivial formalist system. 
+
+\secc \sysname{Order}
+
+\sysrules{A \term{sentence} is a member of a finite set of integers. 
+
+\hi Sentence $Y$ is \term{implied by} sentence $X$ if and only if $Y=X$, or else of all the sentences, $Y$ is the one next smaller than $X$. 
+
+\hi Take as the \term{axiom} the largest sentence.}
+
+Is the pure syntax of \sysname{Illusions} isomorphic to \sysname{Order}? The preceding paragraph proved that it is not. The implication structure of \sysname{Order} is mechanical to the point of idiocy, while the implication structure of \sysname{Illusions} is, as I pointed out, elusive. Figure \ref{orderstruc} where loops indicate multiple occurances of the same sentence, could adequately represent a proof in \dq{Order,} but could not remotely represent one in \sysname{Illusions.} The essence of \sysname{Illusions} is that it is coupled to the reader's subjectivity. For an ink-on-paper system even to be comparable to \sysname{Illusions,} the subjectivity would have to be moved out of the reader and onto the paper. This is utterly impossible. 
+
+\begin{figure}
+	{\centering \includegraphics[width=4.5in]{img/orderstructure} \par}
+	\label{orderstruc}
+\end{figure}
+
+\midinsert
+{\leftskip=0pt plus1fil\rightskip=0pt plus1fil
+\picw=4.5in\inspic{orderstructure.png} \cskip
+	\caption/f[orderstruc] Implication structure of \sysname{Order}.
+\par}
+\endinsert
+
+Here is the next system. 
+
+\secc \sysname{Innperseqs}}
+
+\sysrules{
+\e{Explanation}: Consider the rainbow halo which appears to surround a small bright light when one looks at it through fogged glass (such as eyeglasses which have been breathed on). The halo consists of concentric circular bands of color. As the fog evaporates, the halo uniformly contracts toward the light. The halo has a vague outer ring, which contracts as the halo does. Of concern here is what happens on one contracting radius of the halo, and specifically what happens on the segment of that radius lying in the vague outer ring: the outer segment. 
+
+\hi A \term{sentence} (or \term{halopoint}) is the changing halo color at a fixed point, in space, in the halo; until the halo contracts past the point. 
+
+\hi Several sentences \term{imply} another sentence if and only if, at some instant, the several sentences are on an outer segment, and the other sentence is the inner endpoint of that outer segment. 
+
+\hi An \term{axiom} is a sentence which is in the initial vague outer ring (before it contracts), and which is not an inner endpoint. 
+
+An \term{innperseq} is a sequence of sequences of sentences on one radius satisfying the following conditions. 
+\begitems\style n
+* The members of the first sequence are axioms, 
+* For each of the other sequences, the first member is implied by the non-first members of the preceding sequence; and the remaining members (if any) are axioms or first members of preceding sequences. 
+* All first members, of sequences other than the last two, appear as non-first members. 
+* No sentence appears as a non-first member more than once. 
+* The last sequence has one member. 
+\enditems}
+
+\newcommand{\innprow}[4]{
+	\parbox{2.25in}{
+		\parbox{0.35in}{\includegraphics[scale=0.25]{img/time#1}}
+		\parbox{0.5in}{#2:}
+		\parbox{1.25in}{
+			\parbox{1.25in}{#3}
+
+			\parbox{1.25in}{#4}}}\vskip 0.5em}
+
+\begin{figure}
+{\centering
+	\parbox{0.15in}{\rotatebox[origin=c]{90}{\
+	{\footnotesize Successive bands represent the vague outer ring at successive times as it fades in toward the small bright light.}}}\begin{minipage}{1.5in}
+		\imgw{1.3in}{img/innperseqs}\vskip 0.1em {\centering\small small bright light \par}
+	\end{minipage}\begin{minipage}{2in}
+\dq{Sentences} at 
+\vskip 1em
+
+	\innprow{1}{$time_1$}{$a_1 a_2 a_3 a_4 a_5 a_6 a_7 b$}{$a_1,a_2 \rightarrow\ b$}
+
+	\innprow{2}{$time_2$}{$a_2 a_3 a_4 a_5 a_6 a_7 b c$}{$a_3 \rightarrow\ c$}
+
+	\innprow{3}{$time_3$}{$a_4 a_5 a_6 a_7 b c d$}{$a_4,a_5 \rightarrow\ d$}
+
+	\innprow{4}{$time_4$}{$a_6 a_7 b c d e$}{$a_6,b \rightarrow\ e$}
+
+	\innprow{5}{$time_5$}{$a_7 b c d e f$}{$a_7,c \rightarrow\ f$}
+
+	\innprow{6}{$time_6$}{$c d e f g$}{$d,e \rightarrow\ g$}
+
+	\vskip 2em
+
+\dq{Axioms} \\
+	\hskip 1em $a_1 a_2 a_3 a_4 a_5 a_6 a_7$ 
+
+	\vskip 2em
+
+Innperseq \\
+$(a_3,a_2,a_1)$ \\
+$(b,a_3)$ \\
+$(c,a_5,a_4)$ \\
+$(d,b,a_6)$ \\
+$(e,c,a_7)$ \\
+$(f,e,d)$ \\
+$(g)$ 
+	\end{minipage}\par}
+	\vskip 1em
+	\caption{Example instance of \sysname{Innperseqs.}}
+	\label{innperdiag}
+\end{figure}
+
+In diagram \ref{innperdiag}, different positions of the vague outer 
+ring at different times are suggested by different shadings. The 
+outer segment moves \dq{down the page.} The figure is by no means 
+an innperseq, but is supposed to help explain the definition. 
+In \sysname{Innperseqs,} a conventional proof would be redundant unless all 
+the statements were on the same radius. And even if the weakest axiom were 
+chosen (the initial outer endpoint), this axiom would imply the initial inner 
+endpoint, and from there the theorem could be reached immediately. In 
+other words, to use the standard definition of \dq{\term{proof}} in \sysname{Innperseqs} 
+would result in an uninteresting derivation structure. Thus, a more 
+interesting derivation structure is defined, the \dq{\term{innperseq.}} The interest of 
+an \dq{\term{innperseq}} is to be as elaborate as the many restrictions in its definition 
+will allow. Proofs are either disregarded in \sysname{Innperseqs}; or else they are 
+identified with innperseqs, and lack Property 8. \sysname{Innperseqs} makes the 
+break with the proof-theorem structure of formalist mathematics. 
+
+Turning to simple implication, an implicand can have many impliors; 
+and there is an infinity of axioms, specified by a general condition. The 
+system has Property 1 in the sense that a sentence can exist at different 
+times and be a member of different implications. It has Property 4 in the 
+sense that the sentences in a specific implication can exist at different times, 
+and the implication holds as long as the sentences exist. It has Property 3 in 
+that an inner endpoint implies itself. The system also has Properties 5 and 7; 
+and lacks Property 2. But, as before, Properties 6 and 9 are another matter. 
+Given several sentences, it is certainly possible to tell mechanically whether 
+one is implied by the others. But when are you given sentences? If one can 
+think the sentences, then relating them is easy---but it is difficult to think the 
+sentences in the first place, even though they objectively exist. The diagram 
+suggests what to look for, but the actual thinking, the actual sentences are 
+another matter. As for Property 9, when \term{theorems} are identified with last 
+members of innperseqs, I hesitate to say whether a derivation of a given 
+sentence can be constructed mechanically. If a sentence is nearer the center 
+than the axioms are, an innperseq can be constructed for it. Or can it? The 
+answer is contingent. \sysname{Innperseqs} is indeterminate because of the difficulty 
+of thinking the sentences, a difficulty which is defined into the system. It is 
+the mathematician's capabilities at a particular instant which delimit the 
+indeterminacies. Precisely because of the difficulty of thinking sentences, I 
+will give several subvariants of the system. 
+
+\midheading{Indeterminacy}
+
+\begin{sysrules}
+A \dq{\term{totally determinate innperseq}} is an innperseq in which one thinks all the 
+sentences. 
+
+An \dq{\term{implior-indeterminate innperseq}} is an innperseq in which one thinks 
+only each implicand and the outer segment it terminates. 
+
+A \dq{\term{sententially indeterminate innperseq}} is an innperseq in which one thinks 
+only the outer segment, and its inner endpoint, as it progresses inward. 
+\end{sysrules}
+
+
+Let us return to the matter of pure and descriptive syntax. The interest 
+of \sysname{Illusions} and \sysname{Innperseqs} is precisely that their abstract structure 
+cannot be separated from their physical and psychological character, and 
+thus that they are not isomorphic to any conventional ink-on-paper system. I 
+am trying to break through to unheard of, and hopefully significant, modes 
+of implication; to define implication structures (and derivation structures) 
+beyond the reach of past mathematics. 
+
+\clearpage
+\section{Constructed Memory Systems}
+\fancyhead[LE]{\textsc{Post-Formalism in Constructed Memories}} \fancyhead[RO]{\textit{2. Constructed Memory Systems}}
+
+In order to understand this section, it is necessary to be thoroughly 
+familiar with \essaytitle{Studies in Constructed Memories,} the essay following this 
+one. (I have not combined the two essays because their approaches are too 
+different.) I will define post-formalist systems in constructed memories, 
+beginning with a system in an M*-Memory.\ednote{The term M*-Memory is defined on page \pageref{mstardef}.}
+
+\midheading{\dq{Dream Amalgams}}
+
+\sysrules{A \term{sentence} is a possible method, an $A_{a_i}$. with respect to an M*-Memory. 
+The sentence $A_{a_p}$ \dq{\term{implies}} the sentence $A_{a_q}$ if and only if the $a_q$th 
+M*-assertion is actually thought; and either $A_{a_q} = A_{a_p}$, or else there is 
+cross-method contact of a mental state in $A_{a_q}$ with a state in $A_{a_p}$.
+
+The \term{axioms} must be chosen from sentences which satisfy two conditions. 
+The mental states in the sentences must have cross-method contact 
+with mental states in other sentences. And the M*-assertions 
+corresponding to the sentences must not be thought. 
+
+\e{Explanation}: As \essaytitle{Studies in Constructed Memories} says, there can be 
+cross-method contact of states, because a normal dream can 
+combine totally different episodes in the dreamer's life into an 
+amalgam.}
+
+\sysname{Dream Amalgams} has Properties 1--5. For the first time, sentences are 
+structurally composite, with mental states being the relevant sentential 
+elements. Implication has an unusual character. The traditional type of 
+implication, modus ponens, is \dq{directed,} because the conditional is 
+directed. Even if $\ulcorner\varphi\supset\phi\urcorner$ is true 
+$\ulcorner\phi\supset\varphi\urcorner$ may not be. Now implication is also 
+directed in \sysname{Dream Amalgams,} but for a very different reason. 
+Cross-method contact, unlike the conditional, has a symmetric character. 
+What prevents implication from being necessarily symmetrical is that the 
+implicand's M*-assertion actually has to be thought, while the implior's 
+M*-assertion does not. Thus, implication is both subjective and mechanical, 
+it is subjective, in that it is a matter of volition which method is remembered 
+to have actually: been used. It is mechanical, in that when one is 
+remembering, one is automatically aware of the cross-method contacts of 
+states in $A_{a_q}$. The conditions on the axioms ensure that they will have 
+implications without losing Property 7. 
+
+As for compound implication in \sysname{Dream Amalgams,} the organism 
+with the M*-Memory can't be aware of it at all; because it can't be aware 
+that at different times it remembered different methods to be the one 
+actually used. (In fact, the organism cannot be aware that the system has 
+Property 5, for the same reason.) On the other hand, to an outside observer 
+of the M*-Memory, indirect implication is not only thinkable but 
+mechanical. It is not superfluous because cross-method contact of mental 
+states is not necessarily transitive. The outside observer can decide whether a 
+sentence is a theorem by the following mechanical procedure. Check 
+whether the sentence's M*-assertion has acually been thought; if so, check all 
+sentences which imply it to see if any are axioms; if not, check all the 
+sentences which imply the sentences which imply it to see if any are axioms; 
+etc. The number of possible methods is given as finite, so the procedure is 
+certain to terminate. Again, an unprecedented mode of implication has been 
+defined. 
+
+When a post-formalist system is defined in a constructed memory, the 
+discussion and analysis of the system become a consequence of constructed 
+memory theory and an extension of it. Constructed memory theory, which 
+is quite unusual but still more or less employs deductive inference, is used to 
+study post-formalist modes of inference which are anything but deductive. 
+
+To aid in understanding the next system, which involves infalls in a 
+$D$-Memory, here is an 
+
+{ \vskip 1.5em \centering \large \framebox[1.1\width]{\dq{Exercise to be Read Aloud}} \par\vskip 1.5em}
+
+(Read according to a timer, reading the first word at 0' 0", and prolonging 
+and spacing words so that each sentence ends at the time in parentheses after 
+it. Do not pause netween sentences.) 
+\vskip 1em
+\begin{tabular}{ r p{2.5in} }
+	($event_1$) &  All men are mortal. (17") \\
+
+	($Sentence_1=event_2$) &  The first utterance lasted 17" and ended at 17"; and lasted 15" and ended 1" ago. (59") \\
+
+	($S_2=event_3$) & The second utterance lasted 42" and ended at 59": and lasted 50" and ended 2" ago. (1' 31") \\
+
+	($S_3=event_4$) & The third utterance lasted 32" and ended at 1' 31"; and lasted 40" and ended 1" ago. (2' 16") \\
+\end{tabular}
+\vskip 1em
+
+Since '32' in $S_3$ is greater than '2' in $S_2$, $S_2$ must say that $S_1$ ($=event_2$)
+ended 30" after $S_2$ began, or something equally unclear. The duration of $S_2$
+is greater than the distance into the past to which it refers. This situation is 
+not a real infall, but it should give the reader some intuitive notion of an 
+infall. 
+
+{
+	\clearpage %TODO shitty hacky way to get this unbroken
+\midheading{\dq{Infalls}}
+
+\begin{sysrules}
+	A \term{sentence} is a D-sentence, in a D-Memory such that $event_{j+1}$ is the first 
+thinking of the $j$th D-sentence, for all $j$. 
+
+Two sentences \dq{imply} another if and only if all three are the same; or else 
+the three are adjacent (and can be written $S_{j+1},S_j,S_{j-1}$), and are such 
+that $\delta_j=x_{j+1}-x_j> z_j,$ $S^D_{j-1}$ 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. Diagram \ref{infallsdiag} indicates the relations.) 
+\end{sysrules}
+}
+
+\begin{figure}
+	\centering
+	\includegraphics[width=4in]{img/infallsdiag}
+	\caption{Implication structure of example $D$-Memory infalls.}
+	\label{infallsdiag}
+\end{figure}
+
+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 
+is not explicitly excluded by the definition of a \dq{D-Memory.} If the 
+possibility is granted, then \sysname{Infalls} has Properties 1--5. Direct implication is 
+completely mechanical; it is subjective only in that the involuntary 
+determination of the $z_j$ and other aspects of the memory is a \dq{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. 
+
+\sysname{Dream Amalgams} and \sysname{Infalls} are systems constructed with 
+imaginary elements, systems whose \dq{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 
+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 
+system by a procedure which ensures that constructed memories are 
+involved, but which is otherwise arbitrary? Why not suspend the striving 
+after \dq{interesting} systems, that last vestige of the striving after 
+\dq{correctness,} and see what happens? Why not construct the rules of a 
+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}
+
+\begin{sysrules}
+A \term{sentence} is a(n) \blankspace.
+
+Two sentences \term{imply} a third if and only if the two sentences \blankspace\ the third. 
+
+An \term{axiom} is a sentence that \blankspace.
+\end{sysrules}
+
+
+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 
+is with this series that I will fill in the blanks in the rule schema. In the next 
+stage, I fill the first (second, third) blank with the ceries of expressions 
+preceding the-first (second, third) period in the entire series. 
+
+\midheading{\sysname{Haphazard System}}
+
+\begin{sysrules}
+	A \term{sentence} is a the duration $D$-sentences $\triangle\ (\mathparagraph^m)$ conclude these 
+\dq{$\Phi^*$-Reflec\-tion,} 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 \term{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 $\Phi^*$-Memory, the future interval defined 
+in dream the third. 
+
+An \term{axiom} is a sentence that internally D-sentences, just as the 
+\dq{$\Phi^*$-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{\sysname{Fantasied Amnesia}}
+
+\begin{sysrules}
+A \term{sentence} is a duration or the future force of conviction for the Situation 
+or this system given Situation 1. 
+
+Two sentences \term{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 \term{axiom} is a sentence that internally just sentences $A_{a_2}$.
+\end{sysrules}
+
+It becomes clear in thinking about \sysname{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 
+studying it in the usual manner, is another. Let us take \sysname{Fantasied Amnesia}
+at face value. As one becomes used to its rules, they become somewhat more 
+meaningful. I will say that an \dq{interpretation} of a haphazard system is an 
+explanation of its rules that makes some sense out of what may seem 
+senseless. \dq{Interpreting} is somewhat like finding the conditions for the 
+existence of a constructed memory which seemingly cannot exist. The first 
+rule of \sysname{Fantasied Amnesia} is a disjunction of three substantives. The 
+\dq{Situation} referred to in the second substantive expression is either 
+Situation 1 or else an unspecified situation. The third substantive expression 
+apparently means \dq{this system, assuming Situation 1,} and refers to 
+\sysname{Fantasied Amnesia} itself. The definition of \dq{sentence} is thus meaningful, 
+but very bizarre. The second rule speaks of \dq{the acceptance} as if it were a 
+written assent. The rule then speaks of a \dq{malleable study} as \dq{fantasying} 
+something. This construction is quite weird, but let us try to accept it. The 
+third rule speaks of a sentence that \dq{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 
+\dq{the future interval} in the implication rule of \sysname{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 \sysname{Fantasied 
+Amnesia} has them. For whatever it is worth, \sysname{Fantasied Amnesia} is on 
+balance incomparable to formalist systems. 
+
+My transformation rule schema has the form of a biconditional, in 
+which the right-hand clause is the operative one. If a transformation rule were to 
+vary, in such a way that it could be replaced by a constant rule whose right-hand
+clause was the disjunction of the various right clauses for the variable rule, 
+then the latter would vary \dq{trivially.} I will say that a system whose 
+transformation rule can vary non-trivially is a \dq{heterodeterminate} system. 
+Since I have constructed a haphazard metamathematics, why not a 
+heterodeterminate metamathematics? Consider a mathematician with an 
+$M$-Memory, such that each $A_{a_i}$. is the consistent use of a different 
+transformation rule, a different definition of \dq{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 
+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 \dq{chooses} which of the possible methods is remembered to 
+have actually been used. This situation amounts to a heterodeterminate 
+system. In fact, the metamathematics cannot even be written out this time; I 
+can only describe it metametamathematically in terms of an imaginary 
+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 \sysname{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.\ednote{The appendix contains a presentation of this work.} The preceding remark is the 
+metametamathematical description, or definition, of the system. 
+
+\secc Epilogue
+% \fancyhead[LE]{\textsc{Post-Formalism in Constructed Memories}} \fancyhead[RO]{\textit{3. 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 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 in science. But now that mathematics is wide open, and anything goes, we should be aware more than ever that scientific applicability is the only objective value that mathematics has. I would not have set down constructed memory theory and the post-formalist systems if I did not believe that they could be applied. When and how they will be is another matter. 
+
+And what about the \dq{validity} of formalism? The rise of the formalist position is certainly understandable. The formalists had a commendable, rationalistic desire to eliminate the metaphysical problems associated with mathematics. Moreover, formalism helped stimulate the development of the logic needed in computer technology (and also to stimulate this paper). In spite of the productiveness of the formalist position, however, it now seems beyond dispute that formalism has failed to achieve its original goals. (My pure philosophical writings are the last word on this issue.) Perhaps the main lesson to be learned from the history of formalism is that an idea does not have to be \dq{true} to be productive. 
+
+\nonum\notoc\secc Note
+
+Early versions of \sysname{Illusions} and \sysname{Innperseqs} appeared in my essay \essaytitle{Concept Art,} published in \booktitle{An Anthology}, ed. La Monte Young, New York, 1963. An early, July 1961 version of \sysname{System Such That No One Knows What's Going On} appeared in \journaltitle{dimension 14}, Ann Arbor, 1963, published by the University of Michigan College of Architecture and Design. 
\ No newline at end of file
diff --git a/essays/post_formalism_memories.tex b/essays/post_formalism_memories.tex
deleted file mode 100644
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--- a/essays/post_formalism_memories.tex
+++ /dev/null
@@ -1,717 +0,0 @@
-\newcommand{\midheading}[1]{
-	{ \vskip 1em \centering \large \textsc{#1} \par \vskip 1em }}
-
-
-\chapter{Post-Formalism in Constructed Memories}
- 
-\section{Post-Formalist Mathematics}
-
-\fancyhead{} \fancyfoot{} \fancyfoot[LE,RO]{\thepage}
-\fancyhead[LE]{\textsc{Post-Formalism in Constructed Memories}} \fancyhead[RO]{\textit{Post-Formalist Mathematics}}
-
-Over the last hundred years, a philosophy of pure mathematics has 
-grown up which I prefer to call \enquote{formalism.} As Willard Quine says in the 
-fourth section of his essay "Carnap and Logical Truth,' formalism was 
-inspired by a series of developments which began with non-Euclidian 
-geometry. Quine himself is opposed to formalism, but the formalists have 
-found encouragement in Quine's own book, \booktitle{Mathematical Logic}. The best 
-presentation of the formalist position can be found in Rudolph Carnap's 
-\booktitle{The Logical Syntax of Language}. As a motivation to the reader, and 
-as a heuristic aid, I will relate my study to these two standard books. (It will 
-heip if the reader is thoroughly familiar with them.) it is not important 
-whether Carnap, or Quine, or formalism---or my interpretation of them---is 
-\enquote{correct,} for this essay is neither history nor philosophy. I am using history 
-as a bridge, to give the reader access to some extreme mathematical 
-innovations. 
-
-The formalist position goes as follows. Pure mathematics is the 
-manipulation of the meaningless and arbitrary, but typographically 
-well-defined ink-shapes on paper 
-`$w$,' `$x$,' `$y$,' `$z$,' 
-`\texttt{'},' 
-`$($,' `$)$,' 
-`$\downarrow$,' and `$\in$.' 
-These shapes are manipulated according to arbitrary but well-defined 
-mechanical rules. Actually, the rules mimic the structure of primitive 
-systems such as Euclid's geometry. There are formation rules, mechanical 
-definitions of which concatenations of shapes are \enquote{\term{sentences}.} One sentence 
-is `$((x) (x\in x) \downarrow (x) (x\in x))$.' There are transformation rules, rules for the 
-mechanical derivation of sentences from other sentences. The best known 
-transformation rule is the rule that $\psi$ may be concluded from $\varphi$ and 
-$\ulcorner \varphi \supset \psi \urcorner$; 
-where `$\supset$' is the truth-functional conditional. For later convenience, I will 
-say that $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$ are \enquote{\term{impliors},} 
-and that $\psi$ is the \enquote{\term{implicand}.} 
-Some sentences are designated as \enquote{\term{axioms}.} A \enquote{\term{proof}} is a series of 
-sentences such that each is an axiom or an implicand of preceding sentences. 
-The last sentence in a proof is a \enquote{\term{theorem}.} 
-
-This account is ultra-simplified and non-rigorous, but it is adequate for 
-my purposes. (The reader may have noticed a terminological issue here. For 
-Quine, an implication is merely a logically true conditional. The rules which 
-are used to go from some statements to others, and to assemble proofs, are 
-rules of inference. The relevant rule of inference is the \term{modus ponens};\editornote{i.e., "$P$ implies $Q$. $P$ is true. Therefore, $Q$ must also be true."} $\psi$ is 
-the ponential of $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$. What I 
-am doing is to use a terminology of 
-implication to talk about rules of inference and ponentials. The reason is 
-that the use of Quine's terminology would result in extremely awkward 
-formulations. What I will be doing is sufficiently transparent that it can be 
-translated into Quine's terminology if necessary. My results will be 
-unaffected.) The decisive feature of the arbitrary game called \enquote{mathematics} 
-is as follows. A sentence-series can be mechanically checked to determine 
-whether it is a proof. But there is no mechanical method for deciding 
-whether a sentence is a theorem. Theorems, or rather their proofs, have to be 
-puzzled out, to be discovered. in this feature lies the dynamism, the 
-excitement of traditional mathematics. Traditional mathematical ability is 
-the ability to make inferential discoveries. 
-
-A variety of branches of mathematics can be specialized out from the 
-basic system. Depending on the choices of axioms, systems can be 
-constructed which are internally consistent, but conflict with each other. A 
-system can be \enquote{interpreted,} or given a meaning within the language of a 
-science such as physics. So interpreted, it may have scientific value, or it may 
-not. But as pure mathematics, all the systems have the same arbitrary status. 
-
-By \enquote{formalist mathematics} I will mean the present mathematical 
-systems which are presented along the above lines. Actually, as many authors 
-have observed, the success of the non-Euclidian \enquote{imaginary} geometries 
-made recognition of the game-like character of mathematics inevitable. 
-Formalism is potentially the greatest break with tradition in the history of 
-mathematics. In the \essaytitle{Foreward} to \booktitle{The Logical Syntax of Language}, Carnap 
-brilliantly points out that mathematical innovation is still hindered by the 
-widespread opinion that deviations from mathematical tradition must be 
-justified---that is, proved to be \enquote{correct} and to be a faithful rendering of 
-\enquote{the true logic.} According to Carnap, we are free to choose the rules of a 
-mathematical system arbitrarily. The striving after correctness must cease, so 
-that mathematics will no longer be hindered. \enquote{\emph{Before us lies the boundless 
-ocean of unlimited possibilities.}} In other words, Carnap, the most reputable 
-of academicians, says you can do anything in mathematics. Do not worry 
-whether whether your arbitrary game corresponds to truth, tradition, or 
-reality: it is still legitimate mathematics. Despite this wonderful \uline{Principle of 
-Tolerance} in mathematics, Carnap never ventured beyond the old 
-ink-on-paper, axiomatic-deductive structures. I, however, have taken Carnap 
-at his word. The result is my \enquote{post-formalist mathematics.} I want to stress 
-that my innovations have been legitimized in advance by one of the most 
-reputable academic figures of the twentieth century. 
-
-Early in 1961, I constructed some systems which went beyond 
-formalist mathematics in two respects. 
-\begin{enumerate}[label=\arabic*.,nosep,itemsep=0.5em]
-	\item My sentential elements are physically different from the little ink-shapes on paper used in all formalist systems. My sentences are physically different from concatenations of ink-shapes. My transformation rules have nothing to do with operations on ink-shapes. 
-
-\item My systems do not necessarily follow the axiomatic-deductive, 
-sen\-ten\-ce-implication-axiom-proof-theorem structure. 
-\end{enumerate}
-\vskip 0.5em
-
-		Both of these 
-possibilities, by the way, are mentioned by Carnap in \essaytitle{Languages as 
-Calculi.}\editornote{Also in \booktitle{The Logical Syntax of Language}.} A \enquote{post-formalist system,} then, is a formalist system which differs 
-physically from an ink-on-paper system, or which lacks the 
-axiomatic-deductive structure. 
-
-As a basis for the analysis of post-formalist systems, a list of structural 
-properties of formalist systems is desirable. Here is such a list. By 
-\enquote{implication} I will mean simple, direct implication, unless I say otherwise. 
-\vskip 0.5em
-\begin{enumerate}[nosep, itemsep=0.5em]
-\item A sentence can be repeated at will. 
-
-\item The rule of implication refers to elements of sentences: sentences 
-are structurally composite. 
-
-\item A sentence can imply itself. 
-
-\item The repeat of an implior can imply the repeat of an implicand: an 
-implication can be repeated. 
-
-\item Different impliors can imply different implicands. 
-
-\item Given two or three sentences, it is possible to recognize 
-mechanically whether one or two directly imply the third. 
-
-\item No axiom is implied by other, different axioms. 
-
-\item The definition of \enquote{proof} is the standard definition, in terms of 
-implication, given early in this essay. 
-
-\item Given the axioms and some other sentence, it is not possible to 
-recognize mechanically whether the sentence is a theorem.
-Compound indirect implication is a puzzle. 
-\end{enumerate}
-\vskip 0.5em
-
-Now for the first post-formalist system. 
-
-\midheading{\sysname{Illusions}}
-
-\begin{sysrules}
-A \term{sentence} is the page (page \pageref{illusions}, with figure \ref{illusions} on it) so long as the 
-apparent, perceived ratio of the length of the vertical line to that 
-of the horizontal line (the statement's \enquote{associated ratio}) does not 
-change. (Two sentences are the \enquote{same} if end only if their 
-associated ratios are the same.) 
-
-A sentence $Y$ is \term{implied by} a sentence $X$ if and only if $Y$ is the same as $X$, 
-or else $Y$ is, of all the sentences one ever sees, the sentence having 
-the associated ratio next smaller than that of $X$. 
-
-Take as the \term{axiom} the first sentence one sees. 
-
-\emph{Explanation}: The figure is an optical illusion such that the vertical line 
-normally appears longer than the horizontal line, even though their 
-lengths are equal. One can correct one's perception, come to see 
-the vertical line as shorter relative to the horizontal line, decrease 
-the associated ratio, by measuring the lines with a ruler to convince 
-oneself that the vertical line is not longer than the other, and then 
-trying to see the lines as equal in length; constructing similar 
-figures with a variety of real (measured) ratios and practicing 
-judging these ratios; and so forth. 
-\end{sysrules}
-
-\begin{figure}[p]
-	{\centering \includegraphics[width=4in]{img/illusions} \par}
-	\caption{The sentence for \sysname{Illusions}.}
-	\label{illusions}
-\end{figure}
-
-\sysname{Illusions} has Properties 1, 3--5, and 7--8. Purely to clarify this fact, the 
-following sequence of integers is presented as a model of the order in which 
-associated ratios might appear in reality. (The sequence is otherwise totally 
-inadequate as a model of \sysname{Illusions.}) $4\medspace2\medspace1$; $4\medspace2$; $5\medspace4\medspace2\medspace1$; $4\medspace3\medspace1$. The 
-implication structure would then be as shown in figure \ref{illusionstructure}.
-
-\begin{figure}
-	{\centering \includegraphics[width=4.5in]{img/illusionstructure} \par}
-	\caption{Example implication structure for \sysname{Illusions}.}
-	\label{illusionstructure}
-\end{figure}
-
-The axiom would be 4, and 5 could not appear in a proof. \sysname{Illusions} has 
-Property 1 on the basis that one can control the associated ratio. Turning to 
-Property 4, it is normally the case that when an implication is repeated, a 
-given occurrence of one of the sentences involved is unique to a specific 
-occurrence of the implication. In \sysname{Illusions,} however, if two equal 
-sentences are next smaller than $X$, the occurrence of $X$ does not uniquely 
-belong to either of the two occurrences of the implication. Compare figure \ref{thestructure},
-where the occurrence of `$t$' is not unique to either occurrence of `$the$'. 
-Subject to this explanation, \sysname{Illusions} has Property 4. \sysname{Illusions} has 
-Property 8, but it goes without saying that the type of implication is not 
-\term{modus ponens}. Properties 3, 5, and 7 need no comment. As for Property 2, 
-the rule of implication refers to a property of sentences, rather than to 
-elements of sentences. The interesting feature of \sysname{Illusions} is that it 
-reverses the situation defined by Properties 6 and 9. Compound indirect 
-implication is about the same as simple implication. The only difference is 
-the difference between being smaller and being next smaller. And there is 
-only one axiom (per person). 
-
-\begin{figure}
-	{\centering \setlength\tabcolsep{0.25em}
-	\begin{tabular}{c c c} t & h & e \\ h &   &   \\ e &   & \end{tabular} \par}
-	\caption{Structure with shared node.}
-	\label{thestructure}
-\end{figure}
-
-
-Simple direct implication, however, is subjective and illusive. It 
-essentially involves changing one's perceptions of an illusion. The change of 
-associated ratios is subjective, elusive, and certainly not numerically 
-measurable. Then, the order in which one sees sentences won't always be 
-their order in the implications and proofs. And even though one is exposed 
-to all the sentences, one may have difficulty distinguishing and remembering 
-them in consciousness. If I see the normal illusion, then manage to get 
-myself to see the lines as being of equal length, I know I have seen a 
-theorem. What is difficult is grasping the steps in between, the simple direct 
-implications. If the brain contains a permanent impression of every sensation 
-it has received, then the implications objectively exist; but they may not be 
-thinkable without neurological techniques for getting at the impressions. In 
-any case, \enquote{proof} is well-defined in some sense---but proofs may not be 
-thinkable. \sysname{Illusions} is, after all, not so much shakier in this respect than 
-even simple arithmetic, which contains undecidable sentences and 
-indefinable terms. 
-
-In \booktitle{The Logical Syntax of Language}, Carnap distinguishes pure syntax 
-and descriptive syntax; and says that pure syntax should be independent of 
-notation, and that every system should be isomorphic to some ink-on-paper 
-system. In so doing, Carnap violates his own \uline{Principle of Tolerance}. Consider 
-the following trivial formalist system. 
-
-\midheading{\enquote{Order}}
-
-\begin{sysrules}
-A \term{sentence} is a member of a finite set of integers. 
-
-Sentence $Y$ is \term{implied by} sentence $X$ if and only if $Y=X$, or else of all the 
-sentences, $Y$ is the one next smaller than $X$. 
-
-Take as the \term{axiom} the largest sentence. 
-\end{sysrules}
-
-Is the pure syntax of \sysname{Illusions} isomorphic to \sysname{Order}? The preceding 
-paragraph proved that it is not. The implication structure of \sysname{Order} is 
-mechanical to the point of idiocy, while the implication structure of 
-\sysname{Illusions} is, as I pointed out, elusive. Figure \ref{orderstruc}
-where loops indicate multiple occurances of the same sentence, could 
-adequately represent a proof in \enquote{Order,} but could not remotely represent 
-one in \sysname{Illusions.} The essence of \sysname{Illusions} is that it is coupled to the 
-reader's subjectivity. For an ink-on-paper system even to be comparable to 
-\sysname{Illusions,} the subjectivity would have to be moved out of the reader and 
-onto the paper. This is utterly impossible. 
-
-\begin{figure}
-	{\centering \includegraphics[width=4.5in]{img/orderstructure} \par}
-	\caption{Implication structure of \sysname{Order}.}
-	\label{orderstruc}
-\end{figure}
-
-Here is the next system. 
-
-\midheading{\sysname{Innperseqs}}
-
-\begin{sysrules}
-\emph{Explanation}: Consider the rainbow halo which appears to surround a small 
-bright light when one looks at it through fogged glass (such as 
-eyeglasses which have been breathed on). The halo consists of 
-concentric circular bands of color. As the fog evaporates, the halo 
-uniformly contracts toward the light. The halo has a vague outer 
-ring, which contracts as the halo does. Of concern here is what 
-happens on one contracting radius of the halo, and specifically 
-what happens on the segment of that radius lying in the vague 
-outer ring: the outer segment. 
-
-A \term{sentence} (or \term{halopoint}) is the changing halo color at a fixed point, in 
-space, in the halo; until the halo contracts past the point. 
-
-Several sentences \term{imply} another sentence if and only if, at some instant, 
-the several sentences are on an outer segment, and the other 
-sentence is the inner endpoint of that outer segment. 
-
-An \term{axiom} is a sentence which is in the initial vague outer ring (before it 
-contracts), and which is not an inner endpoint. 
-
-An \term{innperseq} is a sequence of sequences of sentences on one radius 
-satisfying the following conditions. 
-	\begin{enumerate}
-		\item The members of the first sequence are axioms, 
-		\item For each of the other sequences, the first member is implied by the non-first members of the preceding sequence; and the remaining members (if any) are axioms or first members of preceding sequences. 
-		\item All first members, of sequences other than the last two, appear as non-first members. 
-		\item No sentence appears as a non-first member more than once. 
-		\item The last sequence has one member. 
-	\end{enumerate}
-\end{sysrules}
-
-\newcommand{\innprow}[4]{
-	\parbox{2.25in}{
-		\parbox{0.35in}{\includegraphics[scale=0.25]{img/time#1}}
-		\parbox{0.5in}{#2:}
-		\parbox{1.25in}{
-			\parbox{1.25in}{#3}
-
-			\parbox{1.25in}{#4}}}\vskip 0.5em}
-
-\begin{figure}
-{\centering
-	\parbox{0.15in}{\rotatebox[origin=c]{90}{\
-	{\footnotesize Successive bands represent the vague outer ring at successive times as it fades in toward the small bright light.}}}\begin{minipage}{1.5in}
-		\imgw{1.3in}{img/innperseqs}\vskip 0.1em {\centering\small small bright light \par}
-	\end{minipage}\begin{minipage}{2in}
-\enquote{Sentences} at 
-\vskip 1em
-
-	\innprow{1}{$time_1$}{$a_1 a_2 a_3 a_4 a_5 a_6 a_7 b$}{$a_1,a_2 \rightarrow\ b$}
-
-	\innprow{2}{$time_2$}{$a_2 a_3 a_4 a_5 a_6 a_7 b c$}{$a_3 \rightarrow\ c$}
-
-	\innprow{3}{$time_3$}{$a_4 a_5 a_6 a_7 b c d$}{$a_4,a_5 \rightarrow\ d$}
-
-	\innprow{4}{$time_4$}{$a_6 a_7 b c d e$}{$a_6,b \rightarrow\ e$}
-
-	\innprow{5}{$time_5$}{$a_7 b c d e f$}{$a_7,c \rightarrow\ f$}
-
-	\innprow{6}{$time_6$}{$c d e f g$}{$d,e \rightarrow\ g$}
-
-	\vskip 2em
-
-\enquote{Axioms} \\
-	\hskip 1em $a_1 a_2 a_3 a_4 a_5 a_6 a_7$ 
-
-	\vskip 2em
-
-Innperseq \\
-$(a_3,a_2,a_1)$ \\
-$(b,a_3)$ \\
-$(c,a_5,a_4)$ \\
-$(d,b,a_6)$ \\
-$(e,c,a_7)$ \\
-$(f,e,d)$ \\
-$(g)$ 
-	\end{minipage}\par}
-	\vskip 1em
-	\caption{Example instance of \sysname{Innperseqs.}}
-	\label{innperdiag}
-\end{figure}
-
-In diagram \ref{innperdiag}, different positions of the vague outer 
-ring at different times are suggested by different shadings. The 
-outer segment moves \enquote{down the page.} The figure is by no means 
-an innperseq, but is supposed to help explain the definition. 
-In \sysname{Innperseqs,} a conventional proof would be redundant unless all 
-the statements were on the same radius. And even if the weakest axiom were 
-chosen (the initial outer endpoint), this axiom would imply the initial inner 
-endpoint, and from there the theorem could be reached immediately. In 
-other words, to use the standard definition of \enquote{\term{proof}} in \sysname{Innperseqs} 
-would result in an uninteresting derivation structure. Thus, a more 
-interesting derivation structure is defined, the \enquote{\term{innperseq.}} The interest of 
-an \enquote{\term{innperseq}} is to be as elaborate as the many restrictions in its definition 
-will allow. Proofs are either disregarded in \sysname{Innperseqs}; or else they are 
-identified with innperseqs, and lack Property 8. \sysname{Innperseqs} makes the 
-break with the proof-theorem structure of formalist mathematics. 
-
-Turning to simple implication, an implicand can have many impliors; 
-and there is an infinity of axioms, specified by a general condition. The 
-system has Property 1 in the sense that a sentence can exist at different 
-times and be a member of different implications. It has Property 4 in the 
-sense that the sentences in a specific implication can exist at different times, 
-and the implication holds as long as the sentences exist. It has Property 3 in 
-that an inner endpoint implies itself. The system also has Properties 5 and 7; 
-and lacks Property 2. But, as before, Properties 6 and 9 are another matter. 
-Given several sentences, it is certainly possible to tell mechanically whether 
-one is implied by the others. But when are you given sentences? If one can 
-think the sentences, then relating them is easy---but it is difficult to think the 
-sentences in the first place, even though they objectively exist. The diagram 
-suggests what to look for, but the actual thinking, the actual sentences are 
-another matter. As for Property 9, when \term{theorems} are identified with last 
-members of innperseqs, I hesitate to say whether a derivation of a given 
-sentence can be constructed mechanically. If a sentence is nearer the center 
-than the axioms are, an innperseq can be constructed for it. Or can it? The 
-answer is contingent. \sysname{Innperseqs} is indeterminate because of the difficulty 
-of thinking the sentences, a difficulty which is defined into the system. It is 
-the mathematician's capabilities at a particular instant which delimit the 
-indeterminacies. Precisely because of the difficulty of thinking sentences, I 
-will give several subvariants of the system. 
-
-\midheading{Indeterminacy}
-
-\begin{sysrules}
-A \enquote{\term{totally determinate innperseq}} is an innperseq in which one thinks all the 
-sentences. 
-
-An \enquote{\term{implior-indeterminate innperseq}} is an innperseq in which one thinks 
-only each implicand and the outer segment it terminates. 
-
-A \enquote{\term{sententially indeterminate innperseq}} is an innperseq in which one thinks 
-only the outer segment, and its inner endpoint, as it progresses inward. 
-\end{sysrules}
-
-
-Let us return to the matter of pure and descriptive syntax. The interest 
-of \sysname{Illusions} and \sysname{Innperseqs} is precisely that their abstract structure 
-cannot be separated from their physical and psychological character, and 
-thus that they are not isomorphic to any conventional ink-on-paper system. I 
-am trying to break through to unheard of, and hopefully significant, modes 
-of implication; to define implication structures (and derivation structures) 
-beyond the reach of past mathematics. 
-
-\clearpage
-\section{Constructed Memory Systems}
-\fancyhead[LE]{\textsc{Post-Formalism in Constructed Memories}} \fancyhead[RO]{\textit{2. Constructed Memory Systems}}
-
-In order to understand this section, it is necessary to be thoroughly 
-familiar with \essaytitle{Studies in Constructed Memories,} the essay following this 
-one. (I have not combined the two essays because their approaches are too 
-different.) I will define post-formalist systems in constructed memories, 
-beginning with a system in an M*-Memory.\editornote{The term M*-Memory is defined on page \pageref{mstardef}.}
-
-\midheading{\enquote{Dream Amalgams}}
-
-\begin{sysrules}
-A \term{sentence} is a possible method, an $A_{a_i}$. with respect to an M*-Memory. 
-The sentence $A_{a_p}$ \enquote{\term{implies}} the sentence $A_{a_q}$ if and only if the $a_q$th 
-M*-assertion is actually thought; and either $A_{a_q} = A_{a_p}$, or else there is 
-cross-method contact of a mental state in $A_{a_q}$ with a state in $A_{a_p}$.
-
-The \term{axioms} must be chosen from sentences which satisfy two conditions. 
-The mental states in the sentences must have cross-method contact 
-with mental states in other sentences. And the M*-assertions 
-corresponding to the sentences must not be thought. 
-
-\emph{Explanation}: As \essaytitle{Studies in Constructed Memories} says, there can be 
-cross-method contact of states, because a normal dream can 
-combine totally different episodes in the dreamer's life into an 
-amalgam. 
-\end{sysrules}
-
-\sysname{Dream Amalgams} has Properties 1--5. For the first time, sentences are 
-structurally composite, with mental states being the relevant sentential 
-elements. Implication has an unusual character. The traditional type of 
-implication, modus ponens, is \enquote{directed,} because the conditional is 
-directed. Even if $\ulcorner\varphi\supset\phi\urcorner$ is true 
-$\ulcorner\phi\supset\varphi\urcorner$ may not be. Now implication is also 
-directed in \sysname{Dream Amalgams,} but for a very different reason. 
-Cross-method contact, unlike the conditional, has a symmetric character. 
-What prevents implication from being necessarily symmetrical is that the 
-implicand's M*-assertion actually has to be thought, while the implior's 
-M*-assertion does not. Thus, implication is both subjective and mechanical, 
-it is subjective, in that it is a matter of volition which method is remembered 
-to have actually: been used. It is mechanical, in that when one is 
-remembering, one is automatically aware of the cross-method contacts of 
-states in $A_{a_q}$. The conditions on the axioms ensure that they will have 
-implications without losing Property 7. 
-
-As for compound implication in \sysname{Dream Amalgams,} the organism 
-with the M*-Memory can't be aware of it at all; because it can't be aware 
-that at different times it remembered different methods to be the one 
-actually used. (In fact, the organism cannot be aware that the system has 
-Property 5, for the same reason.) On the other hand, to an outside observer 
-of the M*-Memory, indirect implication is not only thinkable but 
-mechanical. It is not superfluous because cross-method contact of mental 
-states is not necessarily transitive. The outside observer can decide whether a 
-sentence is a theorem by the following mechanical procedure. Check 
-whether the sentence's M*-assertion has acually been thought; if so, check all 
-sentences which imply it to see if any are axioms; if not, check all the 
-sentences which imply the sentences which imply it to see if any are axioms; 
-etc. The number of possible methods is given as finite, so the procedure is 
-certain to terminate. Again, an unprecedented mode of implication has been 
-defined. 
-
-When a post-formalist system is defined in a constructed memory, the 
-discussion and analysis of the system become a consequence of constructed 
-memory theory and an extension of it. Constructed memory theory, which 
-is quite unusual but still more or less employs deductive inference, is used to 
-study post-formalist modes of inference which are anything but deductive. 
-
-To aid in understanding the next system, which involves infalls in a 
-$D$-Memory, here is an 
-
-{ \vskip 1.5em \centering \large \framebox[1.1\width]{\enquote{Exercise to be Read Aloud}} \par\vskip 1.5em}
-
-(Read according to a timer, reading the first word at 0' 0", and prolonging 
-and spacing words so that each sentence ends at the time in parentheses after 
-it. Do not pause netween sentences.) 
-\vskip 1em
-\begin{tabular}{ r p{2.5in} }
-	($event_1$) &  All men are mortal. (17") \\
-
-	($Sentence_1=event_2$) &  The first utterance lasted 17" and ended at 17"; and lasted 15" and ended 1" ago. (59") \\
-
-	($S_2=event_3$) & The second utterance lasted 42" and ended at 59": and lasted 50" and ended 2" ago. (1' 31") \\
-
-	($S_3=event_4$) & The third utterance lasted 32" and ended at 1' 31"; and lasted 40" and ended 1" ago. (2' 16") \\
-\end{tabular}
-\vskip 1em
-
-Since '32' in $S_3$ is greater than '2' in $S_2$, $S_2$ must say that $S_1$ ($=event_2$)
-ended 30" after $S_2$ began, or something equally unclear. The duration of $S_2$
-is greater than the distance into the past to which it refers. This situation is 
-not a real infall, but it should give the reader some intuitive notion of an 
-infall. 
-
-{
-	\clearpage %TODO shitty hacky way to get this unbroken
-\midheading{\enquote{Infalls}}
-
-\begin{sysrules}
-	A \term{sentence} is a D-sentence, in a D-Memory such that $event_{j+1}$ is the first 
-thinking of the $j$th D-sentence, for all $j$. 
-
-Two sentences \enquote{imply} another if and only if all three are the same; or else 
-the three are adjacent (and can be written $S_{j+1},S_j,S_{j-1}$), and are such 
-that $\delta_j=x_{j+1}-x_j> z_j,$ $S^D_{j-1}$ 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. Diagram \ref{infallsdiag} indicates the relations.) 
-\end{sysrules}
-}
-
-\begin{figure}
-	\centering
-	\includegraphics[width=4in]{img/infallsdiag}
-	\caption{Implication structure of example $D$-Memory infalls.}
-	\label{infallsdiag}
-\end{figure}
-
-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 
-is not explicitly excluded by the definition of a \enquote{D-Memory.} If the 
-possibility is granted, then \sysname{Infalls} has Properties 1--5. Direct implication is 
-completely mechanical; it is subjective only in that the involuntary 
-determination of the $z_j$ and other aspects of the memory is a \enquote{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. 
-
-\sysname{Dream Amalgams} and \sysname{Infalls} are systems constructed with 
-imaginary elements, systems whose \enquote{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 
-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 
-system by a procedure which ensures that constructed memories are 
-involved, but which is otherwise arbitrary? Why not suspend the striving 
-after \enquote{interesting} systems, that last vestige of the striving after 
-\enquote{correctness,} and see what happens? Why not construct the rules of a 
-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}
-
-\begin{sysrules}
-A \term{sentence} is a(n) \blankspace.
-
-Two sentences \term{imply} a third if and only if the two sentences \blankspace\ the third. 
-
-An \term{axiom} is a sentence that \blankspace.
-\end{sysrules}
-
-
-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 
-is with this series that I will fill in the blanks in the rule schema. In the next 
-stage, I fill the first (second, third) blank with the ceries of expressions 
-preceding the-first (second, third) period in the entire series. 
-
-\midheading{\sysname{Haphazard System}}
-
-\begin{sysrules}
-	A \term{sentence} is a the duration $D$-sentences $\triangle\ (\mathparagraph^m)$ conclude these 
-\enquote{$\Phi^*$-Reflec\-tion,} 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 \term{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 $\Phi^*$-Memory, the future interval defined 
-in dream the third. 
-
-An \term{axiom} is a sentence that internally D-sentences, just as the 
-\enquote{$\Phi^*$-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{\sysname{Fantasied Amnesia}}
-
-\begin{sysrules}
-A \term{sentence} is a duration or the future force of conviction for the Situation 
-or this system given Situation 1. 
-
-Two sentences \term{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 \term{axiom} is a sentence that internally just sentences $A_{a_2}$.
-\end{sysrules}
-
-It becomes clear in thinking about \sysname{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 
-studying it in the usual manner, is another. Let us take \sysname{Fantasied Amnesia}
-at face value. As one becomes used to its rules, they become somewhat more 
-meaningful. I will say that an \enquote{interpretation} of a haphazard system is an 
-explanation of its rules that makes some sense out of what may seem 
-senseless. \enquote{Interpreting} is somewhat like finding the conditions for the 
-existence of a constructed memory which seemingly cannot exist. The first 
-rule of \sysname{Fantasied Amnesia} is a disjunction of three substantives. The 
-\enquote{Situation} referred to in the second substantive expression is either 
-Situation 1 or else an unspecified situation. The third substantive expression 
-apparently means \enquote{this system, assuming Situation 1,} and refers to 
-\sysname{Fantasied Amnesia} itself. The definition of \enquote{sentence} is thus meaningful, 
-but very bizarre. The second rule speaks of \enquote{the acceptance} as if it were a 
-written assent. The rule then speaks of a \enquote{malleable study} as \enquote{fantasying} 
-something. This construction is quite weird, but let us try to accept it. The 
-third rule speaks of a sentence that \enquote{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 
-\enquote{the future interval} in the implication rule of \sysname{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 \sysname{Fantasied 
-Amnesia} has them. For whatever it is worth, \sysname{Fantasied Amnesia} is on 
-balance incomparable to formalist systems. 
-
-My transformation rule schema has the form of a biconditional, in 
-which the right-hand clause is the operative one. If a transformation rule were to 
-vary, in such a way that it could be replaced by a constant rule whose right-hand
-clause was the disjunction of the various right clauses for the variable rule, 
-then the latter would vary \enquote{trivially.} I will say that a system whose 
-transformation rule can vary non-trivially is a \enquote{heterodeterminate} system. 
-Since I have constructed a haphazard metamathematics, why not a 
-heterodeterminate metamathematics? Consider a mathematician with an 
-$M$-Memory, such that each $A_{a_i}$. is the consistent use of a different 
-transformation rule, a different definition of \enquote{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 
-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 \enquote{chooses} which of the possible methods is remembered to 
-have actually been used. This situation amounts to a heterodeterminate 
-system. In fact, the metamathematics cannot even be written out this time; I 
-can only describe it metametamathematically in terms of an imaginary 
-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 \sysname{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.\editornote{The appendix contains a presentation of this work.} The preceding remark is the 
-metametamathematical description, or definition, of the system. 
-
-\section{Epilogue}
-\fancyhead[LE]{\textsc{Post-Formalism in Constructed Memories}} \fancyhead[RO]{\textit{3. 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 
-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 
-in science. But now that mathematics is wide open, and anything goes, we 
-should be aware more than ever that scientific applicability is the only 
-objective value that mathematics has. I would not have set down constructed 
-memory theory and the post-formalist systems if I did not believe that they 
-could be applied. When and how they will be is another matter. 
-
-And what about the \enquote{validity} of formalism? The rise of the formalist 
-position is certainly understandable. The formalists had a commendable, 
-rationalistic desire to eliminate the metaphysical problems associated with 
-mathematics. Moreover, formalism helped stimulate the development of the 
-logic needed in computer technology (and also to stimulate this paper). In 
-spite of the productiveness of the formalist position, however, it now seems 
-beyond dispute that formalism has failed to achieve its original goals. (My 
-pure philosophical writings are the last word on this issue.) Perhaps the main 
-lesson to be learned from the history of formalism is that an idea does not 
-have to be \enquote{true} to be productive. 
-
-
-\section*{Note}
-Early versions of \sysname{Illusions} and \sysname{Innperseqs} appeared in my essay 
-\essaytitle{Concept Art,} published in \booktitle{An Anthology}, ed. La Monte Young, New 
-York, 1963. An early, July 1961 version of \sysname{System Such That No One 
-Knows What's Going On} appeared in \journaltitle{dimension 14}, Ann Arbor, 1963, 
-published by the University of Michigan College of Architecture and Design. 
-
diff --git a/essays/studies_in_constructed_memories.tex b/essays/studies_in_constructed_memories.otx
index 9c370df..9c370df 100644
--- a/essays/studies_in_constructed_memories.tex
+++ b/essays/studies_in_constructed_memories.otx