From 8548929f5fcdc5dde833337247b223ce614a8199 Mon Sep 17 00:00:00 2001
From: grr <grr@lo2.org>
Date: Thu, 2 May 2024 20:51:35 -0400
Subject: para-science breakout

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+\chapter{Post-Formalism in Constructed Memories}
+\section{Post-Formalist Mathematics}
+
+Over the last hundred years, a philosophy of pure mathematics has 
+grown up which I prefer to call "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 
+"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,' '{}',' '(,' '),' '$\downarrow$,' and '$\in$.' 
+These shapes are manipulated according to arbitrary but well-detined 
+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 "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 
+trasformation 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 "impliors," 
+and that $\psi$ is the "implicand." 
+Some sentences are designated as "axioms." A "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 "theorem." 
+
+This account is ultrasimplified 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 modus ponens; $\psi$ 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 "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 "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 "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 "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 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 "correct" and to be a faithful rendering of 
+"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. \said{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 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 "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. 1. 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. 2. My systems do not necessarily follow the axiomatic-deductive, 
+sentence-implication-axiom-proof-theorem structure. Both of these 
+possibilities, by the way, are mentioned by Carnap in \papertitle{Languages as 
+Calculi.} A "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 
+"implication" I will mean simple, direct implication, unless I say otherwise. 
+\begin{enumerate}
+\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 "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}
+
+Now for the first post-formalist system. 
+
+{ \centering \large "\textsc{Illusions}" \par}
+
+\begin{sysrules}
+A "sentence" is the following page (with the figure 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 "associated ratio") does not 
+change. (Two sentences are the "same" if end only if their 
+associated ratios are the same.) 
+
+A sentence Y is "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 axiom the first sentence one sees. 
+
+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}
+
+\img{illusions}
+
+"IIlusions" 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 "Illusions.") 4 2 1; 4 2; 5 4 2 1; 4 3 1. The 
+implication structure would then be 
+
+\img{illusionstructure}
+
+The axiom would be 4, and 5 could not appear in a proof. "IIlusions" 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 "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 '\begin{tabular}{c c c} t & h & e \\ h &   &   \\ e &   & \end{tabular}', 
+where the occurrence of 't' is not unique to either occurrence of 'the'. 
+Subject to this explanation, "Illusions" has Property 4. "Illusions" has 
+Property 8, but it goes without saying that the type of implication is not 
+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 "IIlusions" 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). 
+
+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, "proof" is well-defined in some sense---but proofs may not be 
+thinkable. "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 ov'n Principle of Tolerance. Consider 
+the following trivial formalist system. 
+
+{ \centering \large "\textsc{Order}" \par}
+
+\begin{sysrules}
+A "sentence" is a member of a finite set of integers. 
+
+Sentence Y is "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 axiom the largest sentence. 
+\end{sysrules}
+
+Is the pure syntax of "\textsc{Illusions}" insomorphic to "\textsc{Order}"? The preceding 
+paragraph proved that it is not. The implication structure of "Order" is 
+mechanical to the point of idiocy, while the implication structure of 
+"Illusions" is, as I pointed out, elusive. The figure 
+
+\img{orderstructure}
+
+where loops indicate multiple occurances of the same sentence, could 
+adequately represent a proof in "Order," but could not remotely represent 
+one in "Illusions." The essence of "Illusions" is that it is coupled to the 
+reader's subjectivity. For an ink-on-paper system even to be comparable to 
+"IIlusions," the subjectivity would have to be moved out of the reader and 
+onto the paper. This is utterly impossible. 
+
+Here is the next system. 
+
+{ \centering \large "\textsc{Innperseqs}" \par}
+
+\begin{sysrules}
+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 "sentence" (or halopoint) is the changing halo color at a fixed point, in 
+space, in the halo; until the halo contracts past the point. 
+
+Several sentences "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 "axiom" is a sentence which is in the initial vague outer ring (before it 
+contracts), and which is not an inner endpoint. 
+
+An "innperseq" is a sequence of sequences of sentences on one radius 
+satisfying the following conditions. 1. The members of the first 
+sequence are axioms, 2. For each of the other sequences, the first 
+member is implied by the non-first members of the preceding 
+sequence; and the remaining inembers (if any) are axioms or first 
+members of preceding sequences. 3. All first members, of 
+sequences other than the last two, appear as non-first members. 4. 
+No sentence appears as a non-first member more than once. 5. The 
+last sequence has one member. 
+
+In the diagram on the following page, different positions of the vague outer 
+ring at different times are suggested by different shadings. The 
+outer segment moves "down the page." The figure is by no means 
+an innperseq, but is supposed to help explain the definition. 
+\end{sysrules}
+
+Successive bands represent the vague outer ring at successive times as it fades in 
+toward the small bright light.
+
+Innperseqs Diagram 
+
+\img{innperseqs}
+
+"Sentences" at 
+
+	\begin{tabular}{ c r l }
+		\bimg{time1} & $time_1$: & $a_1 a_2 a_3 a_4 a_5 a_6 a_7 b$ \\
+		& & $a_1,a_2 \rightarrow\ b$ \\
+	\end{tabular}
+
+	\begin{tabular}{c r l}
+		\bimg{time2} & $time_2$: & $a_2 a_3 a_4 a_5 a_6 a_7 b c$ \\
+		& & $a_3 \rightarrow\ c$ \\
+	\end{tabular}
+
+	\begin{tabular}{c r l}
+		\bimg{time3} & $time_3$: & $a_4 a_5 a_6 a_7 b c d$ \\
+		& & $a_4,a_5 \rightarrow\ d$ \\
+	\end{tabular}
+
+	\begin{tabular}{c r l}
+		\bimg{time4} & $time_4$: & $a_6 a_7 b c d e$ \\
+		& & $a_6,b \rightarrow\ e$ \\
+	\end{tabular}
+
+	\begin{tabular}{c r l}
+		\bimg{time5} & $time_5$: & $a_7 b c d e f$ \\
+		& & $a_7,c \rightarrow\ f$ \\
+	\end{tabular}
+
+	\begin{tabular}{c r l}
+		\bimg{time6} & $time_6$: & $c d e f g$ \\
+		& & $d,e \rightarrow\ g$ \\
+	\end{tabular}
+
+"Axioms" $a_1 a_2 a_3 a_4 a_5 a_6 a_7$
+
+
+Innperseq \\
+$(a_3,a_2,a_1)$
+$(b,a_3)$
+$(c,a_5,a_4)$
+$(d,b,a_6)$
+$(e,c,a_7)$
+$(f,e,d)$
+$(g)$
+
+In "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 "proof" in "Innperseqs" 
+would result in an uninteresting derivation structure. Thus, a more 
+interesting derivation structure is defined, the "innperseq." The interest of 
+an "innperseq" is to be as elaborate as the many restrictions in its definition 
+will allow. Proofs are either disregarded in "Innperseqs"; or else they are 
+identified with innpersegs, and lack Property 8. "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 "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. "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. 
+
+{ \centering \large \textsc{Indeterminacy} \par}
+\begin{sysrules}
+A "totally determinate innperseq" is an innperseq in which one thinks all the 
+sentences. 
+
+An "implior-indeterminate innperseq" is an innperseq in which one thinks 
+only each implicand and the outer segment it terminates. 
+
+A "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 "Illusions" and "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. 
+
+\subsection{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. 
+
+{ \centering \large "\textsc{Dream Amalgams}" \par}
+
+\begin{sysrules}
+A "sentence" is a possible method, an $A_{a_i}$. with respect to an M*-Memory. 
+The sentence $A_{a_p}$ "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 $A_{q_p}$\footnote{sic?}
+
+The 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. 
+
+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}
+
+"\textsc{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 "directed," because the conditional is 
+directed. Even if $\ulcorner\varphi\supset\phi\urcorner$ is true 
+$\ulcorner\varphi\supset\phi\urcorner$ may not be. Now implication is also 
+directed in "\textsc{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 "\textsc{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 
+
+{ \centering \large \framebox[1.1\width]{"Exercise to be Read Aloud"} \par}
+
+(Read according to a timer, reading the first word at O' O", and prolonging 
+and spacing words so that each sentence ends at the time in parentheses after 
+it. Do not pause netween sentences.) 
+
+\begin{tabular}{ r l }
+	($event_1$) &  All men are mortal. (17") \\
+
+	($Sentence_1=event_2s$) &  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}
+
+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. 
+
+
+\newcommand{\midheading}[1]{
+	{ \centering \large \textsc{#1} \par}}
+
+\midheading{"Infalls"}
+
+\begin{sysrules}
+	A "sentence" is a D-sentence, in a D-Memory such that $event_{j+1}$ is the first 
+thinking of the jth D-sentence, for all j. 
+
+Two sentences "imply" another if and only if all three are the same; or else 
+the three are adjacent (and can be written $S_{j+1},S_j,S_{j-1}$), 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. The diagram indicates the relations.) 
+\end{sysrules}
+
+\img{infallsdiag}
+
+In this variety of D-Memory, the organism continuously thinks successive 
+D-sentences, which are all different, just as the reader of the above exercise 
+continuously reads successive and different sentences. Thus, the possibility 
+of repeating a sentence depends on the possibility of thinking it while one is 
+thinking another sentence---a possibility which may be far-fetched, but which 
+is not explicitly excluded by the definition of a "D-Memory." If the 
+possibility is granted, then "\textsc{Infalls}" has Properties 1--5. Direct implication is 
+completely mechanical; it is subjective only in that the involuntary 
+determination of the $z_j$ and other aspects of the memory is a 'subjective' 
+process of the organism. Compound implication is also mechanical to an 
+outside observer of the memory, but if the organism itself is to be aware of 
+it, it has to perform fantastic feats of multiple thinking. 
+
+"\textsc{Dream Amalgams}" and "\textsc{Infalls}" are systems constructed with 
+imaginary elements, systems whose "notation" is drawn from an imaginary 
+object or system. Such systems have no descriptive syntax. Imaginary objects 
+were introduced into mathematics, or at least into geometry, by Nicholas 
+Lobachevski, and now I am using them as a notation. For these systems to 
+be nonisomorphic to any ink-on-paper systems, the mathematician must be 
+the organism with the M*-Memory or the D*-Memory. But this means that 
+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 "interesting" systems, that last vestige of the striving after 
+"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 "sentence" is a(n) \blankspace.
+
+Two sentences "imply" a third if and only if the two sentences \blankspace\ the third. 
+
+An "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{"Haphazard System"}
+
+\begin{sysrules}
+A "sentence" is a the duration D-sentences $\triangle\ (\mathparagraph^m)$ conclude these 
+"$\Phi*$-Reflection," or the future Assumption voluntarily force of 
+conviction for conclusion the Situation or by ongoing that this 
+system? be given telling between the Situation 1. 
+
+Two sentences "imply" a third if and only if the two sentences is\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 "axiom" is a sentence that internally D-sentences, just as the 
+"$\Phi*$-Memory" sentences $A_{a_1}$ is $A_{a_2}$. 
+
+In the final stage, I cancel the smallest number of words I have to in 
+order to make the rules grammatical. 
+\end{sysrules}
+
+\midheading{"Fantasied Amnesia"}
+
+\begin{sysrules}
+A "sentence" is a duration or the future force of conviction for the Situation 
+or this system given Situation 1. 
+
+Two sentences "imply" a third if and only if the two sentences have the 
+acceptance that a certain and malleable study extra-linguistically can 
+fantasy not remembering the future interval defined in the third. 
+
+An "axiom" is a sentence that internally just sentences $A_{a_2}$.
+\end{sysrules}
+
+It becomes clear in thinking about "Fantasied Amnesia" that its 
+metametamathematics is dual. Describing the construction of the rules, the 
+metamathematics, by a systematic performance, is one thing. Taking the 
+finished metamathematics at face value, independently of its origin, and 
+studying it in the usual manner, is another. Let us take "Fantasied Amnesia" 
+at face value. As one becomes used to its rules, they become somewhat more 
+meaningful. I will say that an "interpretation" of a haphazard system is an 
+explanation of its rules that makes some sense out of what may seem 
+senseless. "Interpreting" is somewhat like finding the conditions for the 
+existence of a constructed memory which seemingly cannot exist. The first 
+rule of "Fantasied Amnesia" is a disjunction of three substantives. The 
+"Situation" referred to in the second substantive expression is either 
+Situation 1 or else an unspecified situation. The third substantive expression 
+apparently means "this system, assuming Situation 1," and refers to 
+"Fantasied Amnesia" itself. The definition of "sentence" is thus meaningful, 
+but very bizarre. The second rule speaks of "the acceptance" as if it were a 
+written assent. The rule then speaks of a "malleable study" as "fantasying" 
+something. This construction is quite weird, but let us try to accept it. The 
+third rule speaks of a sentence that "sentences" (in the legal sense) a possible 
+method. So much for the meaning of the rules. 
+
+Turning to the nine properties of formalist systems, the reference to 
+"the future interval" in the implication rule of "Fantasied Amnesia" 
+indicates that the system has Property 2; and the system can perfectly well 
+have Property 8. It does not have Property 6 in any known sense. Certainly 
+it does have Property 9. it just might have Property. 1. But as for the other 
+four properties, it seems out of the question to decide whether "Fantasied 
+Amnesia" has them. For whatever it is worth, "Fantasied Amnesia" is on 
+balance incomparable to formalist systems. 
+
+My transformation rule schema has the form of a biconditional, in 
+which the right 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 
+clause was the disjunction of the various right clauses for the variable rule, 
+then the latter would vary "trivially." 1 will say that a system whose 
+transformation rule can vary non-trivially is a "heterodeterminate" system. 
+Since 1 have constructed a haphazard metamathematics, why not a 
+heterodeterminate metamathematics? Consider a mathematician with an 
+M-Memory, such that each $A_{a_i}$. is the consistent use of a different 
+transformation rule, a different definition of "imply," for the mathematics 
+in which the mathematician is discovering theorems. The consistent use of a 
+transformation rule is after all a method---a method for finding the 
+commitments premisses make, and for basing conclusions in premisses. When 
+the mathematician goes to remember which rule of inference he has actually 
+been using, he "chooses" which of the possible methods is remembered to 
+have actually been used. This situation amounts to a heterodeterminate 
+system. tn 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 \textsc{"System Such That No One 
+Knows What's Going On."} One just has to guess whether this system exists, 
+and if it does what it is like. The preceding remark is the 
+metametamathematical description, or definition, of the system. 
+
+\subsection{Epilogue}
+
+Ever since Carnap's Principle of Tolerance opened the floodgates to 
+arbitrariness in mathematics, we have been faced with the prospect of a 
+mathematics which is  indistinguishable from  art-for-art's-sake, or 
+amusement-for-amusement's-sake. But there is one characteristic which saves 
+mathematics from this fate. Mathematics originated by abstraction from 
+primitive technology, and is indispensable to science and technology---in 
+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 "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 "true" to be productive. 
+
+
+\section*{Note}
+Early versions of \textsc{"Illusions"} and \textsc{"Innperseqs"} appeared in my essay 
+"Concept Art," published in An Anthology, ed. La Monte Young, New 
+York, 1963. An early, July 1961 version of \textsc{"System Such That No One 
+Knows What's Going On"} appeared in dimension 14, Ann Arbor, 1963, 
+published by the University of Michigan College of Architecture and Design. 
+
-- 
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