From 614bc606467643792652386aa71fe6f006f06282 Mon Sep 17 00:00:00 2001 From: p Date: Wed, 27 Nov 2024 23:07:58 -0500 Subject: mathematical studies up to innperseqs diagram --- aux.otx | 6 + blueprint.otx | 6 +- essays/mathematical_studies.otx | 17 + essays/mathematical_studies.tex | 15 - essays/post_formalism_memories.otx | 518 +++++++++++++++++++++ essays/post_formalism_memories.tex | 717 ----------------------------- essays/studies_in_constructed_memories.otx | 499 ++++++++++++++++++++ essays/studies_in_constructed_memories.tex | 499 -------------------- 8 files changed, 1043 insertions(+), 1234 deletions(-) create mode 100644 essays/mathematical_studies.otx delete mode 100644 essays/mathematical_studies.tex create mode 100644 essays/post_formalism_memories.otx delete mode 100644 essays/post_formalism_memories.tex create mode 100644 essays/studies_in_constructed_memories.otx delete mode 100644 essays/studies_in_constructed_memories.tex diff --git a/aux.otx b/aux.otx index ef7b8db..e93e1f5 100644 --- a/aux.otx +++ b/aux.otx @@ -99,6 +99,12 @@ \sdef{_item:Z}{{\bf\uppercase\_ea{\_athe\_itemnum}.\hskip 0.5em}} \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] 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.otx b/essays/mathematical_studies.otx new file mode 100644 index 0000000..e386199 --- /dev/null +++ b/essays/mathematical_studies.otx @@ -0,0 +1,17 @@ +\chapter 1966 Mathematical Studies + +% \fancyhead{} \fancyfoot{} \fancyfoot[LE,RO]{\thepage} +% \fancyhead[LE]{\textsc{Mathematical Studies (1966)}} \fancyhead[RO]{\textit{Introduction}} +\sec 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 \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. + +The ultimate test of the non-triviality of pure mathematics is whether it has practical applications. I believe that the approaches presented on a very abstract level in this monograph will turn out to have such applications. In order to be applied, the principles which are presented here have to be developed intensively on a level which is compatible with applications. The results will be found in my two subsequent essays, \essaytitle{Subjective Propositional Vibration} and \essaytitle{The Logic of Admissible Contradictions}. + diff --git a/essays/mathematical_studies.tex b/essays/mathematical_studies.tex deleted file mode 100644 index ffb914b..0000000 --- a/essays/mathematical_studies.tex +++ /dev/null @@ -1,15 +0,0 @@ -\chapter{1966 Mathematical Studies: 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. - -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. - -The ultimate test of the non-triviality of pure mathematics is whether it has practical applications. I believe that the approaches presented on a very abstract level in this monograph will turn out to have such applications. In order to be applied, the principles which are presented here have to be developed intensively on a level which is compatible with applications. The results will be found in my two subsequent essays, \essaytitle{Subjective Propositional Vibration} and \essaytitle{The Logic of Admissible Contradictions}. - 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 index 0c867a2..0000000 --- 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.otx b/essays/studies_in_constructed_memories.otx new file mode 100644 index 0000000..9c370df --- /dev/null +++ b/essays/studies_in_constructed_memories.otx @@ -0,0 +1,499 @@ +\chapter{Studies in Constructed Memories} + +\section{Introduction} + +\fancyhead{} \fancyfoot{} \fancyfoot[LE,RO]{\thepage} +\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{1. Introduction}} + +The memory of a conscious organism is a phenomenon in which +interrelations of mind, language, and the rest of reality are especially evident. +In these studies, I will define some conscious memory-systems, and +investigate them. The investigation will be mathematical. In fact, the nearest +precedent for it is perhaps the geometry of Nicholas Lobachevski. +Non-Euclidian geometry had many founders, but Lobachevski in particular +spoke of his system as an \enquote{imaginary geometry.} Lobachevski's system was, +so to speak, the physical geometry of an \enquote{imaginary,} or constructed, space. +By analogy, my investigation could be called a psychological algebra of +constructed minds. It is too early to characterize the investigation more +exactly. Let us just remember Rudoiph Carnap's \uline{Principle of Tolerance} in +mathematics: the mathematician is free to construct his system in any way +he chooses. + +I will begin by introducing a repertory of concepts informally, becoming more formal as I go along. Consider ongoing actions, which by definition extend through past, present, and future. For example, \enquote{I am making the trip from New York to Chicago.} Consider also past actions which have probable consequences in the present. \enquote{I have been heating this water} (entailing that it isn't frozen now). I will be concerned with such actions as these. + +Our language provides for the following assertion: \enquote{I am off to the country today; I could have been off to the beach; I could not possibly have been going to the center of the sun}. We distinguish an actual action from a possible action; and distinguish both from an action which is materially impossible. People insist that there are things they could do, even though they don't choose to do them (as opposed to things they couldn't do). What distinguishes these possible actions from impossible ones? Rather than trying to analyze such everyday notions in terms of the logic of counterfactual conditionals, or of modalities, or of probability, I choose to take the notions at their face value. My concern is not to philosophize, but to assemble concepts with which to define an interesting memory system. + +What is the introspective psychological difference between a thought +that has the force of a memory, and a thought that has the force of a +fantasied past, a merely possible past? I am not asking how I know that a +verbalized memory is true; I am asking what quality a naive thought has that +marks it as a memory. Let Alternative $E$ be that I went to an East Side +restaurant yesterday, and Alternative $W$ be that I went to a West Side one. +By the \enquote{thought of $E$} I mean mainly the visualization of going into the East +Side restaurant. My thought of $E$ has the force of memory. It actually +happened. $W$ is something I could have done. I can imagine I did do W. There +is nothing present which indicates whether I did $E$ or $W$. Yet $W$ merely has +the force of possibility, of fantasy. How do the two thoughts differ? Is the +thought of $E$ involuntarily more vivid? Is there perhaps an \enquote{attitude of +assertion} involuntarily present in the thought of $E$? + +Consider the memory that I was almost run down by a truck yesterday: +I could have been run down, but wasn't. In such a case, the possibility that I +could have been run down would be more vivid than the actuality that I +wasn't. (Is it not insanity, when a person is overwhelmed by the fear of a +merely possible past event?) My hold on sanity here would be the awareness +that I am alive and well today. + +In dreams, do we not wholeheartedly \enquote{remember} that a misfortune +has befallen us, and begin to adjust emotionally to it? Then we awake, and +wholeheartedly remember that the misfortune has not befallen us. The +thought that had the force of memory in the dream ceases to have that force +as we awake. We remember the dream, and conclude that it was a fantasy. +Even more characteristic of dreams, do I not to all intents and purposes go +to far places and carry out all sorts of actions in a dream, only to awaken in +bed? We say that the dream falsifies my present environment, my +sensations, my actions, memories, the past, my whole world, in a totally +convincing way. Can a hypnotist produce artificial dreams, that is, can he +control their content? Can the hypnotist give his subject one false memory +one moment, and replace it with a contradictory memory the next +moment? + +I will now specify a situation involving possible actions and +remembering. + +\newenvironment{hangers} +{\vskip 0.5em\begin{hangparas}{3em}{1}} +{\end{hangparas}\vskip 0.5em} + +\begin{hangers} +\textbf{Situation 1.} \enquote{I could have been accomplishing $G$ by doing $A_{a_1}$, or by +doing $A_{a_2}$, \ldots, or by doing $A_{a_n}$; but I have actually been accomplishing $G$ by +doing $A_{a_1}$.} Here the ongoing actions $A_{a_i}$, $i=1,\ldots,n$,$a_i\neq a_h$ if $i\neq h$, are +the possible methods of accomplishing $G$. (The subscripts are supposed to +indicate that the methods are distinct and countable, but not ordered.) The +possible methods cannot be combined, let us assume. +\end{hangers} + +In such a situation, perhaps the thought that I have been doing $A_{a_1}$ +would be distinguished from similar thoughts about $A_{a_2}, \ldots, A_{a_n}$ by the +presence of the \enquote{attitude of assertion}. Since the possible methods are +ongoing actions, the thought that I have been doing $A_{a_i}$ has logical or +probabie consequences I can check against the present. + +Now $A_{a_1}$, is actual and $A_{a_2}$ is not, so that $A_{a_1}$, simply cannot have +possible jar in $A_{a_3}$ to contain it. The only \enquote{connection} $A_{a_1}$ could have +material contact with $A_{a_2}$. An actual liquid in $A_{a_1}$ could not require a +with $A_{a_2}$, would be verbal and gratuitous. Therefore, in order to be possible +methods, $A_{a_2}$, \ldots, $A_{a_n}$ must be materially separable. A liquid in $A_{a_2}$ must +not require a jar in $A_{a_3}$ to contain it. If it did, $A_{a_2}$ couldn't be actualized +while $A_{a_3}$, remained only a possibility. + +Enough concepts are now at hand for the studies to begin in earnest. + +\section{M-Memories} +\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{2. M-Memories}} + +\newcommand{\definition}{\textbf{Definition.}} +\newcommand{\assumption}[1]{\textit{Assumption #1.}} +\newcommand{\conclusion}[1]{\textbf{Conclusion #1.}} + +\begin{hangers} +\definition\ Given the sentences \enquote{I have actually been doing $A_{a_i}$}, where +the $A_{a_i}$ are non-combinable possible methods as in Situation 1, an +\enquote{M-Memory} is a memory of a conscious organism such that the organism +can think precisely one of the sentences at a time, and any of the sentences +has the force of memory. +\end{hangers} + +This definition refers to language, mind, and the rest of reality in their +interrelations, but the crucial reference is to a property of certain sentences. +I have chosen this formulation precisely because of what I want to +investigate. I want to find the minimal, elegant, extra-linguistic conditions, +whatever they may be, for the existence of an M-Memory (which is defined +by a linguistic property). I can say at once that the conditions must enable +the organism to think the sentences at will, and they must provide that the +memory is consistent with the organism's present awareness. + +\begin{hangers} +\definition\ The \term{P-Memory} of a conscious organism is its conscious +memory of what it did and what happened to it, the past events of its life. I +want to distinguish here the \enquote{personal} memory from the preconscious. + +\definition\ An \term{L-Memory} is a linguistic P-Memory having no +extra-linguistic component. Of course, the linguistic component has +extra-linguistic mental associations which give it \enquote{meaning}--otherwise the +memory wouldn't be conscious. But these associations lack the force of a +mental reliving of the past independent of language. An L-Memory amounts +to extra-linguistic amnesia. + +\assumption{1.1} With respect to normal human memory, when I forget +whether I did $x$, I can't voluntarily give either the thought that I did $x$, or +the thought that I didn't do $x$, the force of memory. I know that I either did +or didn't do $x$, but I can create no conviction for either alternative. (An +introspective observation.) + +\conclusion{1.2} An L-Memory is not sufficient for an M-Memory, even +in the trivial case that the $A_{a_i}$ are beyond perception (as internal bodily +processes are). True, there would be no present perceptions to check the +sentences \enquote{I have actually been doing $A_{a_i}$} against. True, the L-Memory +precludes any extra-linguistic memory-\enquote{feelings} which would conflict with +the sentences. But the L-Memory is otherwise normal. And \textit{Assumption 1.1} +indicates that normally, either precisely one of a number of mutually +exclusive possibilities has the force of memory; or else the organism can give +none of them the force of memory. + +\assumption{1.3} I cannot, from within a natural dream, choose to swith +to another dream. (An introspective observation. A \enquote{natural} dream is a +dream involuntarily produced internally during sleep.) + +\conclusion{1.4} An M-Memory could not be produced by natural +dreaming. It is true that in one dream one sentence could have the force of +memory, and in another dream a different sentence could. But an M-Memory +is such that the organism can choose one sentence-memory one moment and +another the next. See Assumption 1.3. + +\assumption{1.5} Returning to the example of the restaurants, I find +that months after the event, my thought of $E$ no longer has the force of +memory. All I remember now is that I used to remember that I did $E$. I +remember that I did $E$ indirectly, by remembering that I remembered that I +did $E$. (My memory that I did $E$ is becoming an L-Memory.) The assumption +is that a memory of one's remembering can indicate, if not imply, that the +event originally remembered occurred. + +\conclusion{1.6} The following are adequate conditions for the existence +of an M-Memory. +\begin{enumerate} +\item The sentences are the organism's only memory of which +method he has been using. + +\item When the organism thinks \enquote{I have actually been doing $A_{a_i}$}. +then (he artificially dreams that) he has been doing $A_{a_i}$ --- and is +now doing it. + +\item When the dream ends, he does not remember that he +remembered that \enquote{he has been doing $A_{a_i}$,} That is, he does not remember +the dream; and he does not remember that he thought the sentence. These +conditions would permit the existence of an M-Memory or else a memory +indistinguishable to all intents and purposes from an M-Memory. +\end{enumerate} +\end{hangers} + +What I have in mind in \conclusion{1.6} is dreams which are produced +artificially but otherwise have all the remarkable qualities of natural dreams. +There would have to be a state of affairs such that the sentence would +instantly start the dream going. + +So much for the conditions for the existence of an M-Memory. +Consider now what it is like as a mental experience to have an M-Memory. +What present or ongoing awareness accompanies an M-Memory? +\conclusion{1.6.2} already told what the remembering is like. For the rest, I will +informally sketch some conclusions. The organism can extra-linguistically +image the $A_{a_i}$. The organism can think \enquote{I could have been doing $A_{a_i}$.} When +not remembering, the organism doesn't have to do any $A_{a_i}$, or he can do any +one of them. The organism must not do anything which would liquidate a +possble method, render the action no longer possible for him. + +\begin{hangers} +\assumption{2.1} A normal dream can combine two totally different +past episodes in my life into a fused episode, or amalgam; so that I \enquote{relive} it +without doubts as.a single episode, and yet remain vaguely aware that +different episodes are present in it. Dreams have the capacity not only to +falsify my world, but to make the impossible believable. (An introspective +observation.) + +\conclusion{2.2} The conditions for the existence of an M-Memory +further permit material contact between the possible methods, the very +contact which is out of the question in a normal Situation 1. The dream is so +flexible that the organism can dream that an (actual) liquid is\slash was contained +by a jar in a possible method. See \assumption{2.1} Thus, the $A_{a_i}$ do not have +to be separable to be possible methods. +\end{hangers} + +I will now introduce further concepts pertaining to the mind. + +\begin{hangers} +\definition\ A \term{mental state} is a mental \enquote{stage} or \enquote{space} or \enquote{mood} +in which visualizing, remembering, and all imaging can be carried on. +\end{hangers} + +Some human mental states are stupor, general anxiety, empathy with +another person, dizziness, general euphoria, clearheadedness (the normal +state in which work is performed), and dreaming. In all but the last state, +some simple visualization routine could be carried out voluntarily. Even ina +dream, I can have visualizations, although here I can't have them at will. The +states are not defined by the imaging or activities carried on while in them, +but are \enquote{spaces} in which such imaging or activities are carried on. + +By definition. + +\begin{hangers} +\conclusion{3.2} An M-Memory has to occur within the time which the +possible methods require, the time required to accomplish G. By definition. + +\definition\ An \term{M*-Memory} is an M-Memory satisfying these +conditions. \label{mstardef} +\begin{enumerate} +\item $A_{a_i}$, for the entire time it requires, involves the voluntary +assuming of mental states. $i=1,...,n$. +\item The material contact between the +possible methods, the cross-method contact, is specifically some sort of +contact between states. +\end{enumerate} + +\conclusion{3.3} For an M*-Memory, to remember is to choose the +mental state in which the remembering is required to occur (by the +memory). After all, for any M-Memory, to remember is to choose all the +$A_{a_i}$-required things you are doing while you remember. +\end{hangers} + +By now, the character of this investigation should be clearer. I seek to +stretch our concepts, rather that to find the \enquote{true} ones. The investigation +may appear similar to the old discipline of philosophical psychology, but its +thrust is rather toward the modern axiomatic systems. The reasoning is +loose, but not arbitrary. And the investigation will become increasingly +mathematical. + +\section{D-Memories} +\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{3. D-Memories}} + +\begin{hangers} +\definition\ A \term{D-Memory} is a memory such that measured past time + appears in it only in the following sentences: \enquote{$Event_j$ occurred in the interval +% TODO\ ? whats up with AF +of time which is $x_j-x_{j-1}$ long and ended at $x_j$ $AF$, and is $y_j$ long and ended $z_j$ +\ ago,} where $x_j$, $y_j$ and $z_j$ are positive numbers of time units (such as hours) +and \enquote{$AF$} means \enquote{after a fixed beginning time.} $x_O=O;$ $x_j> x_{j-1}$; and at any +one fixed time, the intervals $|z_j, z_j+y_j|$ nowhere overlap. $y_j+z_j\leq x_j$ For an +integer $m$, the $m$th sentence acquires the force of memory, is added to the +memory, at the fixed time $x_m$. $j=1, \ldots, f(t)$, where the number of sentences +$f(t)$ is written as a function of time $AF$. Then $f(t)=m$ when $x_m \leq t \less x_{m+1}$. +The sentences have the force of memory involuntarily. The organism does +not make them up at will. +\end{hangers} + +Let me explain what the D-Memory involves. $Event_j$ is assigned to an +abnormal \enquote{interval,} a dual interval defined in two unrelated ways. The +intervals defined by the $y_j$ and $z_j$ are tied to the present instant rather than to +a fixed time, and could be written $|N-z_j-y_j, N-z_j|$, where '$N$' means "the time +of the present instant relative to the fixed beginning time." + +\newcommand{\proof}{\textit{Proof}} + +\begin{hangers} +\conclusion{4} The intervals $|N-z_j-y_j, N-z_j|$ nowhere overlap. + +\proof: By definition, the intervals $|z_j, z_j+y_j|$ nowhere overlap. If $j\neq k$, +$|z_j, z_j+y_j|\cap|z_k, z_k+y_k|=\emptyset$ +This fact implies that e.g. $z_j\less z_j+y_j\less z_k\less z_k+y_k$. +Then $N-z_k-y_k\less N-z_k\less N-z_j-y_j\less N-z_j$. +Then $|N-z_k-y_k, N-z_k|\cap|N-z_j-y_j, N-z_j|=\emptyset$ +At any one time, the organism can think of all the sliding intervals, and they +partly cover the time up to now without overlapping. +\end{hangers} + +Suppose you find the deck of $n$ cards + +{ \centering {\vskip 1em} + \framebox[1.1\width]{ + \parbox{1in}{ + \centering + \large $event_{j}$ + {\vskip 0.25em} + $z_{j}$ ago + } + } + \par {\vskip 0.5em} +} + + + +($j=1,\ldots,n$ and $z_j$ is a positive number of days), and you have no +information to date them other than what they themselves say. If you +believe the cards, your mental experience will be a little like having a +D-Memory. Then, the definition does not require that $y_j=x_j-x_{j-1}$. Again, it is +not that two concepts of \enquote{length} are involved, but that the \enquote{interval} is +abnormal. Of course this is all inconsistent, but I want to study the +conditions under which a mind will accept inconsistency. + +\begin{hangers} +\assumption{5.1} With respect to normal human memory, it is possible +to forget what day it is, even though one remembers a past date. (An +empirical observation.) + +\assumption{5.2} This assumption is based on the fact that the sign +\enquote{\textsc{Closed for Vacation. Back in two weeks.}} was in the window of +a nearby store for at least a month this summer; and the fact that a +filmmaker wrote in a newspaper, \enquote{When an actor asks me when the film will +be finished, I say \enquote{In two months,} and two months later I give the same +answer, and I'm always right.} Even in normal circumstances, humans can +maintain a dual and outright inconsistent awareness of measured time. In +general, inconsistency is a normal aspect of human thinking and even has +practical value. +\end{hangers} + +Imagine a child who has been told to date events by saying, for +example, $x$ happened two days ago, and a day later saying again, $x$ happened +two days ago---and who has not been told that this is inconsistent. What +conditions are required for the acceptance of this dating system? It is +precisely because of Assumptions 5.1 and 5.2 that a certain answer cannot +be given to this question. The human mind is so flexible and malleable that +there is no telling how much inconsistency it can absorb. I can only study +what flaws might lead the child to reject the system. The child might \enquote{feel} +that an event recedes into the past, something the memory doesn't express. +An event might be placed by the memory no later than another, and yet +\enquote{feel} more recent than the other. I speculate that if anything will discredit +the system, it will be its conflict with naive, \enquote{felt,} extra-linguistic memory. + +\begin{hangers} +\conclusion{5.3} The above dating system would be acceptable to an +organism with an L-Memory. + +\conclusion{5.4} The existence of an L-Memory is an adequate condition +for the existence of a D-Memory. With extra-linguistic amnesia, the +structure of the language would be the structure of the past in any case. The +past would have no form independent of language. Anyway, time is gone for +good, leaving nothing that can be checked directly. Without an +extra-linguistic memory to fall back on, and considering Assumptions 5.1 +and 5.2, the dual temporal memory shouldn't be too much to absorb. +\end{hangers} + +As I said, the real difficulty with this line of investigation is putting +limits on anything so flexible as the mind's capacity to absorb inconsistency. + +Now the thinking of a sentence in a D-Memory itself takes time. Let +$\delta(S^D_j)$ be the minimum number of time units it takes to think the jth +D-sentence. This function, abbreviated '$\delta_j$', is the duration function of the +D-sentences. + +\begin{hangers} +\conclusion{6.1} If $\delta_j\greater z_j$, the memory of the interval defined by $y_j$ and +$z_j$ places the end of the interval after the beginning of the memory of it, or +does something else equally unclear. If $\delta_j\greater y_j+z_j$, the entire interval is placed +after the beginning of the memory of it. When $\delta_j\greater z_j$, let us say that the end +of the remembered interval falis within the interval for the memory of it, or +that the situation is an \enquote{infall.} (Compare \enquote{The light went out a half-second +ago}.) + +\conclusion{6.2} If $\delta_j\greater x_{j+k}-x_j$, then $S^D_{j+k}$ is added to the preconscious + before $S^D_j$ can be thought once. The earliest interval during which the $j^{th}$ + sentence can be thought \enquote{passes over} the $(j+k)^{th}$ interval. Let us say that +the situation is a \enquote{passover.} (Something of the sort is true of humans, +whose brains contain permanent impressions of far more sensations than can +be thought, remembered in consciousness.) + +\conclusion{6.3} If there are passovers in a D-Memory, the organism +cannot both think the sentences during the earliest intervals possible and be +aware of the passovers. + +\proof: The only way the organism can be aware of $\delta(S_j)$ +is for $event_{j+h}$ ($h$ a positive integer) to be the thinking of $S_j$. +If the thinking of $S_j$ takes piace as the $(j+1)^{th}$ event, then the organism gets two +values for $\delta(S_j)$, namely $x_{j+1}-x_j$ and $y_{j+1}$. Assume that only $x_{j+1}-x_j$ +is allowed as a measure of $\delta(S_j)$. Since $\delta(S_j)=x_{j+1}-x_j$, there is no passover. If +the thinking of $S_j$ takes place as the $(j+2)^{th}$ event, then $x_{j+2}-x{j+1}=\delta(S_j)$ +could be greater than $x_{j+1}-x_j$. But since $S_j$ goes into the preconscious at $x_j$, +$S_j$ is not actually thought in the earliest interval during which it could be +thought. See diagram \ref{dmemdiag}. + +\begin{figure} + \centering + \includegraphics[width=4in]{img/dmemdiag} + \caption{tktk} + \label{dmemdiag} +\end{figure} + +\conclusion{6.4} Let there be an \term{infall} in the case where $event_{j+1}$ is the +thinking of $S_j$. $\delta(S_j)=x_{j+1}-x_j$ and $\delta(S_j)\greater z_j$. $S_{j+1}$ gives $\delta(S_j)$, +so that the organism can be aware of it. +It is greater than $z_j$. Thus, the organism can be +aware of the infall. However, the infall will certainly be no more difficult to +accept than the other features of the D-Memory. And the thinking of $S_j$ has +to be one of the events for the organism to be aware of the infall. +\end{hangers} + +\section{$\Phi$-Memories} +\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{4. $\Phi$-Memories}} +I will conclude these studies with two complex constructions. + +\begin{hangers} +\definition\ A \enquote{$\Phi$-Memory} is a memory which includes an M*-Memory +and a D-Memory, with the following conditions. +\begin{enumerate} +\item The goal $G$, for the M*-Memory, is to move from one point to another. + +\item For the D-Memory, \enquote{$event_j$} becomes a numerical term, the decrease in the organism's distance +from the destination point during the temporal interval. \enquote{A 3-inch move +toward the destination} is the sort of thing that \enquote{$event_j$} here refers to. + +\item The number of $A_{a_i}$ equals the number of D-sentences factorial. The number +of D-sentences, of course, increases. +\end{enumerate} +\end{hangers} + +Consider the consecutive thinking of each D-sentence precisely once, in +minimum time, while the number of sentences remains constant. Such a +\enquote{D-paragraph} is a permutation of the D-sentences. Let $\mathparagraph^m$ be a +D-paragraph when the number of sentences equals the integer m. There are +$m!$ $\mathparagraph^m$s. When $f(t)=m=3$, one of the six $\mathparagraph^3$s is $S^D_3 S^D_1 S^D_2$, +thought in +minimum time. Assume that the duration $\triangle$ of a D-paragraph depends only +on the number of D-sentences and the $\delta_j$. We can write + +$$ \triangle(\mathparagraph^m)=\sum_{j=1}^{m} \delta_j $$ + +The permutations of the D-sentences, as well as the D-paragraphs, can be +indexed with the $a_i$, just as the possible methods are. + +\begin{hangers} +\definition\ A \enquote{$\Phi^*$-Memory} is a $\Phi$-Memory in which the order of the +sentences in the $a_i$th $\mathparagraph^m$ has the meaning of \enquote{I have actually been doing $A_{a_i}$} +assigned to it. The order is the indication that $A_{a_i}$ has actually been used; it +is the $a_j$th $M^*$-assertion. \enquote{I have actually been doing $A_{a_i}$} is merely an English +translation, and does not appear in the $\Phi^*$-Memory. + +\conclusion{7} Given a $\Phi^*$-Memory, if one D-sentence is forgotten, not +only will there be a gap in the awareness of when what events occurred; it +will be forgotten which method has actually been used. +\end{hangers} + +This conclusion points toward a study in which deformations of the +memory language are related to deformations of general consciousness. + +\begin{hangers} +\definition\ A \enquote{$\Phi^*$-Reflection,} or reflection in the present of a +$\Phi^*$-Memory, is a collection of assertions about the future, derived from a +$\Phi^*$-Memory, as follows. +\begin{enumerate} + \item There are the sentences \enquote{$Event_j$ will occur in the +interval of time which is $x_j-x_{j-1}$ long, and begins at twice the present time +$AF$, minus $x_j AF$; and which is $y_j$ long and begins $z_j$ from now.} If $event_j$ was + a 3-inch move toward the destination in the \enquote{$\Phi^*$-Memory,} the sentence in the +$\Phi^*$-Reflection says that a 3-inch move will be made in the future temporal +interval. + \item The $a_i$th permutation of the sentences defined in (1) is an +assertion which has the meaning of \enquote{I will do $A_{a_i}$}; and the organism can +think precisely one permutation at a time. The $A_{a_i}$, $x_j$, $y_j$, $z_j$, and the rest are +defined as before (so that in particular the permutations can be indexed with +the $a_i$). +\end{enumerate} +\end{hangers} +\begin{hangers} +\conclusion{8} Given that the $\Phi^*$-Memory's temporal intervals $|x_{j-1}, x_j|$ +are reflected as $|2N-x_j, 2N-x_{j-1}|$, the reflection preserves the intervals' +absolute distances from the present. + +\proof: The least distance of $|x_{j-1}, x_j|$ +from $N$ is $N-x_j$; the greatest distance is $N-x_{j-1}$. Adding the least distance, and +then the greatest distance, to $N$, gives $|2N-x_j, 2N-x_{j-1}|$. +\end{hangers} + +I will end with two problems. If a $\Phi^*$-Memory exists, under what +conditions will a $\Phi^*$-Reflection be a precognition? Under what conditions +will every assertion be prescience or foreknowledge? By a \enquote{precognition} I +don't mean a prediction about the future implied by deterministic laws; I +mean a direct \enquote{memory} of the future unconnected with general principles. + +Finally, what would a precognitive $\Phi^*$-Reflection be like as a mental +experience? What present or ongoing awareness would accompany a +precognitive $\Phi^*$-Reflection? + diff --git a/essays/studies_in_constructed_memories.tex b/essays/studies_in_constructed_memories.tex deleted file mode 100644 index 9c370df..0000000 --- a/essays/studies_in_constructed_memories.tex +++ /dev/null @@ -1,499 +0,0 @@ -\chapter{Studies in Constructed Memories} - -\section{Introduction} - -\fancyhead{} \fancyfoot{} \fancyfoot[LE,RO]{\thepage} -\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{1. Introduction}} - -The memory of a conscious organism is a phenomenon in which -interrelations of mind, language, and the rest of reality are especially evident. -In these studies, I will define some conscious memory-systems, and -investigate them. The investigation will be mathematical. In fact, the nearest -precedent for it is perhaps the geometry of Nicholas Lobachevski. -Non-Euclidian geometry had many founders, but Lobachevski in particular -spoke of his system as an \enquote{imaginary geometry.} Lobachevski's system was, -so to speak, the physical geometry of an \enquote{imaginary,} or constructed, space. -By analogy, my investigation could be called a psychological algebra of -constructed minds. It is too early to characterize the investigation more -exactly. Let us just remember Rudoiph Carnap's \uline{Principle of Tolerance} in -mathematics: the mathematician is free to construct his system in any way -he chooses. - -I will begin by introducing a repertory of concepts informally, becoming more formal as I go along. Consider ongoing actions, which by definition extend through past, present, and future. For example, \enquote{I am making the trip from New York to Chicago.} Consider also past actions which have probable consequences in the present. \enquote{I have been heating this water} (entailing that it isn't frozen now). I will be concerned with such actions as these. - -Our language provides for the following assertion: \enquote{I am off to the country today; I could have been off to the beach; I could not possibly have been going to the center of the sun}. We distinguish an actual action from a possible action; and distinguish both from an action which is materially impossible. People insist that there are things they could do, even though they don't choose to do them (as opposed to things they couldn't do). What distinguishes these possible actions from impossible ones? Rather than trying to analyze such everyday notions in terms of the logic of counterfactual conditionals, or of modalities, or of probability, I choose to take the notions at their face value. My concern is not to philosophize, but to assemble concepts with which to define an interesting memory system. - -What is the introspective psychological difference between a thought -that has the force of a memory, and a thought that has the force of a -fantasied past, a merely possible past? I am not asking how I know that a -verbalized memory is true; I am asking what quality a naive thought has that -marks it as a memory. Let Alternative $E$ be that I went to an East Side -restaurant yesterday, and Alternative $W$ be that I went to a West Side one. -By the \enquote{thought of $E$} I mean mainly the visualization of going into the East -Side restaurant. My thought of $E$ has the force of memory. It actually -happened. $W$ is something I could have done. I can imagine I did do W. There -is nothing present which indicates whether I did $E$ or $W$. Yet $W$ merely has -the force of possibility, of fantasy. How do the two thoughts differ? Is the -thought of $E$ involuntarily more vivid? Is there perhaps an \enquote{attitude of -assertion} involuntarily present in the thought of $E$? - -Consider the memory that I was almost run down by a truck yesterday: -I could have been run down, but wasn't. In such a case, the possibility that I -could have been run down would be more vivid than the actuality that I -wasn't. (Is it not insanity, when a person is overwhelmed by the fear of a -merely possible past event?) My hold on sanity here would be the awareness -that I am alive and well today. - -In dreams, do we not wholeheartedly \enquote{remember} that a misfortune -has befallen us, and begin to adjust emotionally to it? Then we awake, and -wholeheartedly remember that the misfortune has not befallen us. The -thought that had the force of memory in the dream ceases to have that force -as we awake. We remember the dream, and conclude that it was a fantasy. -Even more characteristic of dreams, do I not to all intents and purposes go -to far places and carry out all sorts of actions in a dream, only to awaken in -bed? We say that the dream falsifies my present environment, my -sensations, my actions, memories, the past, my whole world, in a totally -convincing way. Can a hypnotist produce artificial dreams, that is, can he -control their content? Can the hypnotist give his subject one false memory -one moment, and replace it with a contradictory memory the next -moment? - -I will now specify a situation involving possible actions and -remembering. - -\newenvironment{hangers} -{\vskip 0.5em\begin{hangparas}{3em}{1}} -{\end{hangparas}\vskip 0.5em} - -\begin{hangers} -\textbf{Situation 1.} \enquote{I could have been accomplishing $G$ by doing $A_{a_1}$, or by -doing $A_{a_2}$, \ldots, or by doing $A_{a_n}$; but I have actually been accomplishing $G$ by -doing $A_{a_1}$.} Here the ongoing actions $A_{a_i}$, $i=1,\ldots,n$,$a_i\neq a_h$ if $i\neq h$, are -the possible methods of accomplishing $G$. (The subscripts are supposed to -indicate that the methods are distinct and countable, but not ordered.) The -possible methods cannot be combined, let us assume. -\end{hangers} - -In such a situation, perhaps the thought that I have been doing $A_{a_1}$ -would be distinguished from similar thoughts about $A_{a_2}, \ldots, A_{a_n}$ by the -presence of the \enquote{attitude of assertion}. Since the possible methods are -ongoing actions, the thought that I have been doing $A_{a_i}$ has logical or -probabie consequences I can check against the present. - -Now $A_{a_1}$, is actual and $A_{a_2}$ is not, so that $A_{a_1}$, simply cannot have -possible jar in $A_{a_3}$ to contain it. The only \enquote{connection} $A_{a_1}$ could have -material contact with $A_{a_2}$. An actual liquid in $A_{a_1}$ could not require a -with $A_{a_2}$, would be verbal and gratuitous. Therefore, in order to be possible -methods, $A_{a_2}$, \ldots, $A_{a_n}$ must be materially separable. A liquid in $A_{a_2}$ must -not require a jar in $A_{a_3}$ to contain it. If it did, $A_{a_2}$ couldn't be actualized -while $A_{a_3}$, remained only a possibility. - -Enough concepts are now at hand for the studies to begin in earnest. - -\section{M-Memories} -\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{2. M-Memories}} - -\newcommand{\definition}{\textbf{Definition.}} -\newcommand{\assumption}[1]{\textit{Assumption #1.}} -\newcommand{\conclusion}[1]{\textbf{Conclusion #1.}} - -\begin{hangers} -\definition\ Given the sentences \enquote{I have actually been doing $A_{a_i}$}, where -the $A_{a_i}$ are non-combinable possible methods as in Situation 1, an -\enquote{M-Memory} is a memory of a conscious organism such that the organism -can think precisely one of the sentences at a time, and any of the sentences -has the force of memory. -\end{hangers} - -This definition refers to language, mind, and the rest of reality in their -interrelations, but the crucial reference is to a property of certain sentences. -I have chosen this formulation precisely because of what I want to -investigate. I want to find the minimal, elegant, extra-linguistic conditions, -whatever they may be, for the existence of an M-Memory (which is defined -by a linguistic property). I can say at once that the conditions must enable -the organism to think the sentences at will, and they must provide that the -memory is consistent with the organism's present awareness. - -\begin{hangers} -\definition\ The \term{P-Memory} of a conscious organism is its conscious -memory of what it did and what happened to it, the past events of its life. I -want to distinguish here the \enquote{personal} memory from the preconscious. - -\definition\ An \term{L-Memory} is a linguistic P-Memory having no -extra-linguistic component. Of course, the linguistic component has -extra-linguistic mental associations which give it \enquote{meaning}--otherwise the -memory wouldn't be conscious. But these associations lack the force of a -mental reliving of the past independent of language. An L-Memory amounts -to extra-linguistic amnesia. - -\assumption{1.1} With respect to normal human memory, when I forget -whether I did $x$, I can't voluntarily give either the thought that I did $x$, or -the thought that I didn't do $x$, the force of memory. I know that I either did -or didn't do $x$, but I can create no conviction for either alternative. (An -introspective observation.) - -\conclusion{1.2} An L-Memory is not sufficient for an M-Memory, even -in the trivial case that the $A_{a_i}$ are beyond perception (as internal bodily -processes are). True, there would be no present perceptions to check the -sentences \enquote{I have actually been doing $A_{a_i}$} against. True, the L-Memory -precludes any extra-linguistic memory-\enquote{feelings} which would conflict with -the sentences. But the L-Memory is otherwise normal. And \textit{Assumption 1.1} -indicates that normally, either precisely one of a number of mutually -exclusive possibilities has the force of memory; or else the organism can give -none of them the force of memory. - -\assumption{1.3} I cannot, from within a natural dream, choose to swith -to another dream. (An introspective observation. A \enquote{natural} dream is a -dream involuntarily produced internally during sleep.) - -\conclusion{1.4} An M-Memory could not be produced by natural -dreaming. It is true that in one dream one sentence could have the force of -memory, and in another dream a different sentence could. But an M-Memory -is such that the organism can choose one sentence-memory one moment and -another the next. See Assumption 1.3. - -\assumption{1.5} Returning to the example of the restaurants, I find -that months after the event, my thought of $E$ no longer has the force of -memory. All I remember now is that I used to remember that I did $E$. I -remember that I did $E$ indirectly, by remembering that I remembered that I -did $E$. (My memory that I did $E$ is becoming an L-Memory.) The assumption -is that a memory of one's remembering can indicate, if not imply, that the -event originally remembered occurred. - -\conclusion{1.6} The following are adequate conditions for the existence -of an M-Memory. -\begin{enumerate} -\item The sentences are the organism's only memory of which -method he has been using. - -\item When the organism thinks \enquote{I have actually been doing $A_{a_i}$}. -then (he artificially dreams that) he has been doing $A_{a_i}$ --- and is -now doing it. - -\item When the dream ends, he does not remember that he -remembered that \enquote{he has been doing $A_{a_i}$,} That is, he does not remember -the dream; and he does not remember that he thought the sentence. These -conditions would permit the existence of an M-Memory or else a memory -indistinguishable to all intents and purposes from an M-Memory. -\end{enumerate} -\end{hangers} - -What I have in mind in \conclusion{1.6} is dreams which are produced -artificially but otherwise have all the remarkable qualities of natural dreams. -There would have to be a state of affairs such that the sentence would -instantly start the dream going. - -So much for the conditions for the existence of an M-Memory. -Consider now what it is like as a mental experience to have an M-Memory. -What present or ongoing awareness accompanies an M-Memory? -\conclusion{1.6.2} already told what the remembering is like. For the rest, I will -informally sketch some conclusions. The organism can extra-linguistically -image the $A_{a_i}$. The organism can think \enquote{I could have been doing $A_{a_i}$.} When -not remembering, the organism doesn't have to do any $A_{a_i}$, or he can do any -one of them. The organism must not do anything which would liquidate a -possble method, render the action no longer possible for him. - -\begin{hangers} -\assumption{2.1} A normal dream can combine two totally different -past episodes in my life into a fused episode, or amalgam; so that I \enquote{relive} it -without doubts as.a single episode, and yet remain vaguely aware that -different episodes are present in it. Dreams have the capacity not only to -falsify my world, but to make the impossible believable. (An introspective -observation.) - -\conclusion{2.2} The conditions for the existence of an M-Memory -further permit material contact between the possible methods, the very -contact which is out of the question in a normal Situation 1. The dream is so -flexible that the organism can dream that an (actual) liquid is\slash was contained -by a jar in a possible method. See \assumption{2.1} Thus, the $A_{a_i}$ do not have -to be separable to be possible methods. -\end{hangers} - -I will now introduce further concepts pertaining to the mind. - -\begin{hangers} -\definition\ A \term{mental state} is a mental \enquote{stage} or \enquote{space} or \enquote{mood} -in which visualizing, remembering, and all imaging can be carried on. -\end{hangers} - -Some human mental states are stupor, general anxiety, empathy with -another person, dizziness, general euphoria, clearheadedness (the normal -state in which work is performed), and dreaming. In all but the last state, -some simple visualization routine could be carried out voluntarily. Even ina -dream, I can have visualizations, although here I can't have them at will. The -states are not defined by the imaging or activities carried on while in them, -but are \enquote{spaces} in which such imaging or activities are carried on. - -By definition. - -\begin{hangers} -\conclusion{3.2} An M-Memory has to occur within the time which the -possible methods require, the time required to accomplish G. By definition. - -\definition\ An \term{M*-Memory} is an M-Memory satisfying these -conditions. \label{mstardef} -\begin{enumerate} -\item $A_{a_i}$, for the entire time it requires, involves the voluntary -assuming of mental states. $i=1,...,n$. -\item The material contact between the -possible methods, the cross-method contact, is specifically some sort of -contact between states. -\end{enumerate} - -\conclusion{3.3} For an M*-Memory, to remember is to choose the -mental state in which the remembering is required to occur (by the -memory). After all, for any M-Memory, to remember is to choose all the -$A_{a_i}$-required things you are doing while you remember. -\end{hangers} - -By now, the character of this investigation should be clearer. I seek to -stretch our concepts, rather that to find the \enquote{true} ones. The investigation -may appear similar to the old discipline of philosophical psychology, but its -thrust is rather toward the modern axiomatic systems. The reasoning is -loose, but not arbitrary. And the investigation will become increasingly -mathematical. - -\section{D-Memories} -\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{3. D-Memories}} - -\begin{hangers} -\definition\ A \term{D-Memory} is a memory such that measured past time - appears in it only in the following sentences: \enquote{$Event_j$ occurred in the interval -% TODO\ ? whats up with AF -of time which is $x_j-x_{j-1}$ long and ended at $x_j$ $AF$, and is $y_j$ long and ended $z_j$ -\ ago,} where $x_j$, $y_j$ and $z_j$ are positive numbers of time units (such as hours) -and \enquote{$AF$} means \enquote{after a fixed beginning time.} $x_O=O;$ $x_j> x_{j-1}$; and at any -one fixed time, the intervals $|z_j, z_j+y_j|$ nowhere overlap. $y_j+z_j\leq x_j$ For an -integer $m$, the $m$th sentence acquires the force of memory, is added to the -memory, at the fixed time $x_m$. $j=1, \ldots, f(t)$, where the number of sentences -$f(t)$ is written as a function of time $AF$. Then $f(t)=m$ when $x_m \leq t \less x_{m+1}$. -The sentences have the force of memory involuntarily. The organism does -not make them up at will. -\end{hangers} - -Let me explain what the D-Memory involves. $Event_j$ is assigned to an -abnormal \enquote{interval,} a dual interval defined in two unrelated ways. The -intervals defined by the $y_j$ and $z_j$ are tied to the present instant rather than to -a fixed time, and could be written $|N-z_j-y_j, N-z_j|$, where '$N$' means "the time -of the present instant relative to the fixed beginning time." - -\newcommand{\proof}{\textit{Proof}} - -\begin{hangers} -\conclusion{4} The intervals $|N-z_j-y_j, N-z_j|$ nowhere overlap. - -\proof: By definition, the intervals $|z_j, z_j+y_j|$ nowhere overlap. If $j\neq k$, -$|z_j, z_j+y_j|\cap|z_k, z_k+y_k|=\emptyset$ -This fact implies that e.g. $z_j\less z_j+y_j\less z_k\less z_k+y_k$. -Then $N-z_k-y_k\less N-z_k\less N-z_j-y_j\less N-z_j$. -Then $|N-z_k-y_k, N-z_k|\cap|N-z_j-y_j, N-z_j|=\emptyset$ -At any one time, the organism can think of all the sliding intervals, and they -partly cover the time up to now without overlapping. -\end{hangers} - -Suppose you find the deck of $n$ cards - -{ \centering {\vskip 1em} - \framebox[1.1\width]{ - \parbox{1in}{ - \centering - \large $event_{j}$ - {\vskip 0.25em} - $z_{j}$ ago - } - } - \par {\vskip 0.5em} -} - - - -($j=1,\ldots,n$ and $z_j$ is a positive number of days), and you have no -information to date them other than what they themselves say. If you -believe the cards, your mental experience will be a little like having a -D-Memory. Then, the definition does not require that $y_j=x_j-x_{j-1}$. Again, it is -not that two concepts of \enquote{length} are involved, but that the \enquote{interval} is -abnormal. Of course this is all inconsistent, but I want to study the -conditions under which a mind will accept inconsistency. - -\begin{hangers} -\assumption{5.1} With respect to normal human memory, it is possible -to forget what day it is, even though one remembers a past date. (An -empirical observation.) - -\assumption{5.2} This assumption is based on the fact that the sign -\enquote{\textsc{Closed for Vacation. Back in two weeks.}} was in the window of -a nearby store for at least a month this summer; and the fact that a -filmmaker wrote in a newspaper, \enquote{When an actor asks me when the film will -be finished, I say \enquote{In two months,} and two months later I give the same -answer, and I'm always right.} Even in normal circumstances, humans can -maintain a dual and outright inconsistent awareness of measured time. In -general, inconsistency is a normal aspect of human thinking and even has -practical value. -\end{hangers} - -Imagine a child who has been told to date events by saying, for -example, $x$ happened two days ago, and a day later saying again, $x$ happened -two days ago---and who has not been told that this is inconsistent. What -conditions are required for the acceptance of this dating system? It is -precisely because of Assumptions 5.1 and 5.2 that a certain answer cannot -be given to this question. The human mind is so flexible and malleable that -there is no telling how much inconsistency it can absorb. I can only study -what flaws might lead the child to reject the system. The child might \enquote{feel} -that an event recedes into the past, something the memory doesn't express. -An event might be placed by the memory no later than another, and yet -\enquote{feel} more recent than the other. I speculate that if anything will discredit -the system, it will be its conflict with naive, \enquote{felt,} extra-linguistic memory. - -\begin{hangers} -\conclusion{5.3} The above dating system would be acceptable to an -organism with an L-Memory. - -\conclusion{5.4} The existence of an L-Memory is an adequate condition -for the existence of a D-Memory. With extra-linguistic amnesia, the -structure of the language would be the structure of the past in any case. The -past would have no form independent of language. Anyway, time is gone for -good, leaving nothing that can be checked directly. Without an -extra-linguistic memory to fall back on, and considering Assumptions 5.1 -and 5.2, the dual temporal memory shouldn't be too much to absorb. -\end{hangers} - -As I said, the real difficulty with this line of investigation is putting -limits on anything so flexible as the mind's capacity to absorb inconsistency. - -Now the thinking of a sentence in a D-Memory itself takes time. Let -$\delta(S^D_j)$ be the minimum number of time units it takes to think the jth -D-sentence. This function, abbreviated '$\delta_j$', is the duration function of the -D-sentences. - -\begin{hangers} -\conclusion{6.1} If $\delta_j\greater z_j$, the memory of the interval defined by $y_j$ and -$z_j$ places the end of the interval after the beginning of the memory of it, or -does something else equally unclear. If $\delta_j\greater y_j+z_j$, the entire interval is placed -after the beginning of the memory of it. When $\delta_j\greater z_j$, let us say that the end -of the remembered interval falis within the interval for the memory of it, or -that the situation is an \enquote{infall.} (Compare \enquote{The light went out a half-second -ago}.) - -\conclusion{6.2} If $\delta_j\greater x_{j+k}-x_j$, then $S^D_{j+k}$ is added to the preconscious - before $S^D_j$ can be thought once. The earliest interval during which the $j^{th}$ - sentence can be thought \enquote{passes over} the $(j+k)^{th}$ interval. Let us say that -the situation is a \enquote{passover.} (Something of the sort is true of humans, -whose brains contain permanent impressions of far more sensations than can -be thought, remembered in consciousness.) - -\conclusion{6.3} If there are passovers in a D-Memory, the organism -cannot both think the sentences during the earliest intervals possible and be -aware of the passovers. - -\proof: The only way the organism can be aware of $\delta(S_j)$ -is for $event_{j+h}$ ($h$ a positive integer) to be the thinking of $S_j$. -If the thinking of $S_j$ takes piace as the $(j+1)^{th}$ event, then the organism gets two -values for $\delta(S_j)$, namely $x_{j+1}-x_j$ and $y_{j+1}$. Assume that only $x_{j+1}-x_j$ -is allowed as a measure of $\delta(S_j)$. Since $\delta(S_j)=x_{j+1}-x_j$, there is no passover. If -the thinking of $S_j$ takes place as the $(j+2)^{th}$ event, then $x_{j+2}-x{j+1}=\delta(S_j)$ -could be greater than $x_{j+1}-x_j$. But since $S_j$ goes into the preconscious at $x_j$, -$S_j$ is not actually thought in the earliest interval during which it could be -thought. See diagram \ref{dmemdiag}. - -\begin{figure} - \centering - \includegraphics[width=4in]{img/dmemdiag} - \caption{tktk} - \label{dmemdiag} -\end{figure} - -\conclusion{6.4} Let there be an \term{infall} in the case where $event_{j+1}$ is the -thinking of $S_j$. $\delta(S_j)=x_{j+1}-x_j$ and $\delta(S_j)\greater z_j$. $S_{j+1}$ gives $\delta(S_j)$, -so that the organism can be aware of it. -It is greater than $z_j$. Thus, the organism can be -aware of the infall. However, the infall will certainly be no more difficult to -accept than the other features of the D-Memory. And the thinking of $S_j$ has -to be one of the events for the organism to be aware of the infall. -\end{hangers} - -\section{$\Phi$-Memories} -\fancyhead[LE]{\textsc{Studies in Constructed Memories}} \fancyhead[RO]{\textit{4. $\Phi$-Memories}} -I will conclude these studies with two complex constructions. - -\begin{hangers} -\definition\ A \enquote{$\Phi$-Memory} is a memory which includes an M*-Memory -and a D-Memory, with the following conditions. -\begin{enumerate} -\item The goal $G$, for the M*-Memory, is to move from one point to another. - -\item For the D-Memory, \enquote{$event_j$} becomes a numerical term, the decrease in the organism's distance -from the destination point during the temporal interval. \enquote{A 3-inch move -toward the destination} is the sort of thing that \enquote{$event_j$} here refers to. - -\item The number of $A_{a_i}$ equals the number of D-sentences factorial. The number -of D-sentences, of course, increases. -\end{enumerate} -\end{hangers} - -Consider the consecutive thinking of each D-sentence precisely once, in -minimum time, while the number of sentences remains constant. Such a -\enquote{D-paragraph} is a permutation of the D-sentences. Let $\mathparagraph^m$ be a -D-paragraph when the number of sentences equals the integer m. There are -$m!$ $\mathparagraph^m$s. When $f(t)=m=3$, one of the six $\mathparagraph^3$s is $S^D_3 S^D_1 S^D_2$, -thought in -minimum time. Assume that the duration $\triangle$ of a D-paragraph depends only -on the number of D-sentences and the $\delta_j$. We can write - -$$ \triangle(\mathparagraph^m)=\sum_{j=1}^{m} \delta_j $$ - -The permutations of the D-sentences, as well as the D-paragraphs, can be -indexed with the $a_i$, just as the possible methods are. - -\begin{hangers} -\definition\ A \enquote{$\Phi^*$-Memory} is a $\Phi$-Memory in which the order of the -sentences in the $a_i$th $\mathparagraph^m$ has the meaning of \enquote{I have actually been doing $A_{a_i}$} -assigned to it. The order is the indication that $A_{a_i}$ has actually been used; it -is the $a_j$th $M^*$-assertion. \enquote{I have actually been doing $A_{a_i}$} is merely an English -translation, and does not appear in the $\Phi^*$-Memory. - -\conclusion{7} Given a $\Phi^*$-Memory, if one D-sentence is forgotten, not -only will there be a gap in the awareness of when what events occurred; it -will be forgotten which method has actually been used. -\end{hangers} - -This conclusion points toward a study in which deformations of the -memory language are related to deformations of general consciousness. - -\begin{hangers} -\definition\ A \enquote{$\Phi^*$-Reflection,} or reflection in the present of a -$\Phi^*$-Memory, is a collection of assertions about the future, derived from a -$\Phi^*$-Memory, as follows. -\begin{enumerate} - \item There are the sentences \enquote{$Event_j$ will occur in the -interval of time which is $x_j-x_{j-1}$ long, and begins at twice the present time -$AF$, minus $x_j AF$; and which is $y_j$ long and begins $z_j$ from now.} If $event_j$ was - a 3-inch move toward the destination in the \enquote{$\Phi^*$-Memory,} the sentence in the -$\Phi^*$-Reflection says that a 3-inch move will be made in the future temporal -interval. - \item The $a_i$th permutation of the sentences defined in (1) is an -assertion which has the meaning of \enquote{I will do $A_{a_i}$}; and the organism can -think precisely one permutation at a time. The $A_{a_i}$, $x_j$, $y_j$, $z_j$, and the rest are -defined as before (so that in particular the permutations can be indexed with -the $a_i$). -\end{enumerate} -\end{hangers} -\begin{hangers} -\conclusion{8} Given that the $\Phi^*$-Memory's temporal intervals $|x_{j-1}, x_j|$ -are reflected as $|2N-x_j, 2N-x_{j-1}|$, the reflection preserves the intervals' -absolute distances from the present. - -\proof: The least distance of $|x_{j-1}, x_j|$ -from $N$ is $N-x_j$; the greatest distance is $N-x_{j-1}$. Adding the least distance, and -then the greatest distance, to $N$, gives $|2N-x_j, 2N-x_{j-1}|$. -\end{hangers} - -I will end with two problems. If a $\Phi^*$-Memory exists, under what -conditions will a $\Phi^*$-Reflection be a precognition? Under what conditions -will every assertion be prescience or foreknowledge? By a \enquote{precognition} I -don't mean a prediction about the future implied by deterministic laws; I -mean a direct \enquote{memory} of the future unconnected with general principles. - -Finally, what would a precognitive $\Phi^*$-Reflection be like as a mental -experience? What present or ongoing awareness would accompany a -precognitive $\Phi^*$-Reflection? - -- cgit v1.2.3