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 --- essays/post_formalism_memories.tex | 717 ------------------------------------- 1 file changed, 717 deletions(-) delete mode 100644 essays/post_formalism_memories.tex (limited to 'essays/post_formalism_memories.tex') 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. - -- cgit v1.2.3