mathematical studies up to innperseqs diagram

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-\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. 
-