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diff --git a/essays/post_formalism_memories.tex b/essays/post_formalism_memories.tex new file mode 100644 index 0000000..bf7decc --- /dev/null +++ b/essays/post_formalism_memories.tex @@ -0,0 +1,673 @@ +\chapter{Post-Formalism in Constructed Memories} +\section{Post-Formalist Mathematics} + +Over the last hundred years, a philosophy of pure mathematics has +grown up which I prefer to call "formalism." As Willard Quine says in the +fourth section of his essay "Carnap and Logical Truth,' formalism was +inspired by a series of developments which began with non-Euclidian +geometry. Quine himself is opposed to formalism, but the formalists have +found encouragement in Quine's own book, \booktitle{Mathematical Logic}. The best +presentation of the formalist position can be found in Rudolph Carnap's +\booktitle{The Logical Syntax of Language}. As a motivation to the reader, and +as a heuristic aid, I will relate my study to these two standard books. (It will +heip if the reader is thoroughly familiar with them.) it is not important +whether Carnap, or Quine, or formalism---or my interpretation of them---is +"correct," for this essay is neither history nor philosophy. I am using history +as a bridge, to give the reader access to some extreme mathematical +innovations. + +The formalist position goes as follows. Pure mathematics is the +manipulation of the meaningless and arbitrary, but typographically +well-defined ink-shapes on paper 'w,' 'x,' 'y,' 'z,' '{}',' '(,' '),' '$\downarrow$,' and '$\in$.' +These shapes are manipulated according to arbitrary but well-detined +mechanical rules. Actually, the rules mimic the structure of primitive +systems such as Euclid's geometry. There are formation rules, mechanical +definitions of which concatenations of shapes are "sentences." One sentence +is '$((x) (x\in x) \downarrow (x) (x\in x))$.' There are transformation rules, rules for the +mechanical derivation of sentences from other sentences. The best known +trasformation rule is the rule that $\psi$ may be concluded from $\varphi$ and +$\ulcorner \varphi \supset \psi \urcorner$; +where '$\supset$' is the truth-functional conditional. For later convenience, I will +say that $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$ are "impliors," +and that $\psi$ is the "implicand." +Some sentences are designated as "axioms." A "proof" is a series of +sentences such that each is an axiom or an implicand of preceding sentences. +The last sentence in a proof is a "theorem." + +This account is ultrasimplified and non-rigorous, but it is adequate for +my purposes. (The reader may have noticed a terminological issue here. For +Quine, an implication is merely a logically true conditional. The rules which +are used to go from some statements to others, and to assemble proofs, are +rules of inference. The relevant rule of inference is the modus ponens; $\psi$ is +the ponential of $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$. What I +am doing is to use a terminology of +implication to talk about rules of inference and ponentials. The reason is +that the use of Quine's terminology would result in extremely awkward +formulations. What I will be doing is sufficiently transparent that it can be +translated into Quine's terminology if necessary. My results will be +unaffected.) The decisive feature of the arbitrary game called "mathematics" +is as follows. A sentence-series can be mechanically checked to determine +whether it is a proof. But there is no mechanical method for deciding +whether a sentence is a theorem. Theorems, or rather their proofs, have to be +puzzled out, to be discovered. in this feature lies the dynamism, the +excitement of traditional mathematics. Traditional mathematical ability is +the ability to make inferential discoveries. + +A variety of branches of mathematics can be specialized out from the +basic system. Depending on the choices of axioms, systems can be +constructed which are internally consistent, but conflict with each other. A +system can be "interpreted," or given a meaning within the language of a +science such as physics. So interpreted, it may have scientific value, or it may +not. But as pure mathematics, all the systems have the same arbitrary status. + +By "formalist mathematics" I will mean the present mathematical +systems which are presented along the above lines. Actually, as many authors +have observed, the success of the non-Euclidian "imaginary" geometries +made recognition of the game-like character of mathematics inevitable. +Formalism is potentially the greatest break with tradition in the history of +mathematics. In the Foreward to \booktitle{The Logical Syntax of Language}, Carnap +brilliantly points out that mathematical innovation is still hindered by the +widespread opinion that deviations from mathematical tradition must be +justified---that is, proved to be "correct" and to be a faithful rendering of +"the true logic." According to Carnap, we are free to choose the rules of a +mathematical system arbitrarily. The striving after correctness must cease, so +that mathematics will no longer be hindered. \said{Before us lies the boundless +ocean of unlimited possibilities.} In other words, Carnap, the most reputable +of academicians, says you can do anything in mathematics. Do not worry +whether whether your arbitrary game corresponds to truth, tradition, or +reality: it is still legitimate mathematics. Despite this wonderful Principle of +Tolerance in mathematics, Carnap never ventured beyond the old +ink-on-paper, axiomatic-deductive structures. I, however, have taken Carnap +at his word. The result is my "post-formalist mathematics." I want to stress +that my innovations have been legitimized in advance by one of the most +reputable academic figures of the twentieth century. + +Early in 1961, I constructed some systems which went beyond +formalist mathematics in two respects. 1. My sentential elements are +physically different from the little ink-shapes on paper used in all formalist +systems. My sentences are physically different from concatenations of +ink-shapes. My transformation rules have nothing to do with operations on +ink-shapes. 2. My systems do not necessarily follow the axiomatic-deductive, +sentence-implication-axiom-proof-theorem structure. Both of these +possibilities, by the way, are mentioned by Carnap in \papertitle{Languages as +Calculi.} A "post-formalist system," then, is a formalist system which differs +physically from an ink-on-paper system, or which lacks the +axiomatic-deductive structure. + +As a basis for the analysis of post-formalist systems, a list of structural +properties of formalist systems is desirable. Here is such a list. By +"implication" I will mean simple, direct implication, unless I say otherwise. +\begin{enumerate} +\item A sentence can be repeated at will. + +\item The rule of implication refers to elements of sentences: sentences +are structurally composite. + +\item A sentence can imply itself. + +\item The repeat of an implior can imply the repeat of an implicand: an +implication can be repeated. + +\item Different impliors can imply different implicands. + +\item Given two or three sentences, it is possible to recognize +mechanically whether one or two directly imply the third. + +\item No axiom is implied by other, different axioms. + +\item The definition of "proof" is the standard definition, in terms of +implication, given early in this essay. + +\item Given the axioms and some other sentence, it is not possible to +recognize mechanically whether the sentence is a theorem. +Compound indirect implication is a puzzle. +\end{enumerate} + +Now for the first post-formalist system. + +{ \centering \large "\textsc{Illusions}" \par} + +\begin{sysrules} +A "sentence" is the following page (with the figure on it) so long as the +apparent, perceived ratio of the length of the vertical line to that +of the horizontal line (the statement's "associated ratio") does not +change. (Two sentences are the "same" if end only if their +associated ratios are the same.) + +A sentence Y is "implied by" a sentence X if and only if Y is the same as X, +or else Y is, of all the sentences one ever sees, the sentence having +the associated ratio next smaller than that of X. + +Take as the axiom the first sentence one sees. + +Explanation: The figure is an optical illusion such that the vertical line +normally appears longer than the horizontal line, even though their +lengths are equal. One can correct one's perception, come to see +the vertical line as shorter relative to the horizontal line, decrease +the associated ratio, by measuring the lines with a ruler to convince +oneself that the vertical line is not longer than the other, and then +trying to see the lines as equal in length; constructing similar +figures with a variety of real (measured) ratios and practicing +judging these ratios; and so forth. +\end{sysrules} + +\img{illusions} + +"IIlusions" has Properties 1, 3--5, and 7--8. Purely to clarify this fact, the +following sequence of integers is presented as a model of the order in which +associated ratios might appear in reality. (The sequence is otherwise totally +inadequate as a model of "Illusions.") 4 2 1; 4 2; 5 4 2 1; 4 3 1. The +implication structure would then be + +\img{illusionstructure} + +The axiom would be 4, and 5 could not appear in a proof. "IIlusions" has +Property 1 on the basis that one can control the associated ratio. Turning to +Property 4, it is normally the case that when an implication is repeated, a +given occurrence of one of the sentences involved is unique to a specific +occurrence of the implication. In "Illusions," however, if two equal +sentences are next smaller than X, the occurrence of X does not uniquely + belong to either of the two occurrences of the implication. Compare '\begin{tabular}{c c c} t & h & e \\ h & & \\ e & & \end{tabular}', +where the occurrence of 't' is not unique to either occurrence of 'the'. +Subject to this explanation, "Illusions" has Property 4. "Illusions" has +Property 8, but it goes without saying that the type of implication is not +modus ponens. Properties 3, 5, and 7 need no comment. As for Property 2, +the rule of implication refers to a property of sentences, rather than to +elements of sentences. The interesting feature of "IIlusions" is that it +reverses the situation defined by Properties 6 and 9. Compound indirect +implication is about the same as simple implication. The only difference is +the difference between being smaller and being next smaller. And there is +only one axiom (per person). + +Simple direct implication, however, is subjective and illusive. It +essentially involves changing one's perceptions of an illusion. The change of +associated ratios is subjective, elusive, and certainly not numerically +measurable. Then, the order in which one sees sentences won't always be +their order in the implications and proofs. And even though one is exposed +to all the sentences, one may have difficulty distinguishing and remembering +them in consciousness. If I see the normal illusion, then manage to get +myself to see the lines as being of equal length, I know I have seen a +theorem. What is difficult is grasping the steps in between, the simple direct +implications. If the brain contains a permanent impression of every sensation +it has received, then the implications objectively exist; but they may not be +thinkable without neurological techniques for getting at the impressions. In +any case, "proof" is well-defined in some sense---but proofs may not be +thinkable. "Illusions" is, after all, not so much shakier in this respect than +even simple arithmetic, which contains undecidable sentences and +indefinable terms. + +In \booktitle{The Logical Syntax of Language}, Carnap distinguishes pure syntax +and descriptive syntax; and says that pure syntax should be independent of +notation, and that every system should be isomorphic to some ink-on-paper +system. In so doing, Carnap violates his ov'n Principle of Tolerance. Consider +the following trivial formalist system. + +{ \centering \large "\textsc{Order}" \par} + +\begin{sysrules} +A "sentence" is a member of a finite set of integers. + +Sentence Y is "implied by" sentence X if and only if Y=X, or else of all the +sentences, Y is the one next smaller than X. + +Take as the axiom the largest sentence. +\end{sysrules} + +Is the pure syntax of "\textsc{Illusions}" insomorphic to "\textsc{Order}"? The preceding +paragraph proved that it is not. The implication structure of "Order" is +mechanical to the point of idiocy, while the implication structure of +"Illusions" is, as I pointed out, elusive. The figure + +\img{orderstructure} + +where loops indicate multiple occurances of the same sentence, could +adequately represent a proof in "Order," but could not remotely represent +one in "Illusions." The essence of "Illusions" is that it is coupled to the +reader's subjectivity. For an ink-on-paper system even to be comparable to +"IIlusions," the subjectivity would have to be moved out of the reader and +onto the paper. This is utterly impossible. + +Here is the next system. + +{ \centering \large "\textsc{Innperseqs}" \par} + +\begin{sysrules} +Explanation: Consider the rainbow halo which appears to surround a small +bright light when one looks at it through fogged glass (such as +eyeglasses which have been breathed on). The halo consists of +concentric circular bands of color. As the fog evaporates, the halo +uniformly contracts toward the light. The halo has a vague outer +ring, which contracts as the halo does. Of concern here is what +happens on one contracting radius of the halo, and specifically +what happens on the segment of that radius lying in the vague +outer ring: the outer segment. + +A "sentence" (or halopoint) is the changing halo color at a fixed point, in +space, in the halo; until the halo contracts past the point. + +Several sentences "imply" another sentence if and only if, at some instant, +the several sentences are on an outer segment, and the other +sentence is the inner endpoint of that outer segment. + +An "axiom" is a sentence which is in the initial vague outer ring (before it +contracts), and which is not an inner endpoint. + +An "innperseq" is a sequence of sequences of sentences on one radius +satisfying the following conditions. 1. The members of the first +sequence are axioms, 2. For each of the other sequences, the first +member is implied by the non-first members of the preceding +sequence; and the remaining inembers (if any) are axioms or first +members of preceding sequences. 3. All first members, of +sequences other than the last two, appear as non-first members. 4. +No sentence appears as a non-first member more than once. 5. The +last sequence has one member. + +In the diagram on the following page, different positions of the vague outer +ring at different times are suggested by different shadings. The +outer segment moves "down the page." The figure is by no means +an innperseq, but is supposed to help explain the definition. +\end{sysrules} + +Successive bands represent the vague outer ring at successive times as it fades in +toward the small bright light. + +Innperseqs Diagram + +\img{innperseqs} + +"Sentences" at + + \begin{tabular}{ c r l } + \bimg{time1} & $time_1$: & $a_1 a_2 a_3 a_4 a_5 a_6 a_7 b$ \\ + & & $a_1,a_2 \rightarrow\ b$ \\ + \end{tabular} + + \begin{tabular}{c r l} + \bimg{time2} & $time_2$: & $a_2 a_3 a_4 a_5 a_6 a_7 b c$ \\ + & & $a_3 \rightarrow\ c$ \\ + \end{tabular} + + \begin{tabular}{c r l} + \bimg{time3} & $time_3$: & $a_4 a_5 a_6 a_7 b c d$ \\ + & & $a_4,a_5 \rightarrow\ d$ \\ + \end{tabular} + + \begin{tabular}{c r l} + \bimg{time4} & $time_4$: & $a_6 a_7 b c d e$ \\ + & & $a_6,b \rightarrow\ e$ \\ + \end{tabular} + + \begin{tabular}{c r l} + \bimg{time5} & $time_5$: & $a_7 b c d e f$ \\ + & & $a_7,c \rightarrow\ f$ \\ + \end{tabular} + + \begin{tabular}{c r l} + \bimg{time6} & $time_6$: & $c d e f g$ \\ + & & $d,e \rightarrow\ g$ \\ + \end{tabular} + +"Axioms" $a_1 a_2 a_3 a_4 a_5 a_6 a_7$ + + +Innperseq \\ +$(a_3,a_2,a_1)$ +$(b,a_3)$ +$(c,a_5,a_4)$ +$(d,b,a_6)$ +$(e,c,a_7)$ +$(f,e,d)$ +$(g)$ + +In "Innperseqs," a conventional proof would be redundant unless all +the statements were on the same radius. And even if the weakest axiom were +chosen (the initial outer endpoint), this axiom would imply the initial inner +endpoint, and from there the theorem could be reached immediately. In +other words, to use the standard definition of "proof" in "Innperseqs" +would result in an uninteresting derivation structure. Thus, a more +interesting derivation structure is defined, the "innperseq." The interest of +an "innperseq" is to be as elaborate as the many restrictions in its definition +will allow. Proofs are either disregarded in "Innperseqs"; or else they are +identified with innpersegs, and lack Property 8. "Innperseqs" makes the +break with the proof-theorem structure of formalist mathematics. + +Turning to simple implication, an implicand can have many impliors; +and there is an infinity of axioms, specified by a general condition. The +system has Property 1 in the sense that a sentence can exist at different +times and be a member of different implications. It has Property 4 in the +sense that the sentences in a specific implication can exist at different times, +and the implication holds as long as the sentences exist. It has Property 3 in +that an inner endpoint implies itself. The system also has Properties 5 and 7; +and lacks Property 2. But, as before, Properties 6 and 9 are another matter. +Given several sentences, it is certainly possible to tell mechanically whether +one is implied by the others. But when are you given sentences? If one can +think the sentences, then relating them is easy---but it is difficult to think the +sentences in the first place, even though they objectively exist. The diagram +suggests what to look for, but the actual thinking, the actual sentences are +another matter. As for Property 9, when "theorems" are identified with last +members of innperseqs, I hesitate to say whether a derivation of a given +sentence can be constructed mechanically. If a sentence is nearer the center +than the axioms are, an innperseq can be constructed for it. Or can it? The +answer is contingent. "Innperseqs" is indeterminate because of the difficulty +of thinking the sentences, a difficulty which is defined into the system. It is +the mathematician's capabilities at a particular instant which delimit the +indeterminacies. Precisely because of the difficulty of thinking sentences, I +will give several subvariants of the system. + +{ \centering \large \textsc{Indeterminacy} \par} +\begin{sysrules} +A "totally determinate innperseq" is an innperseq in which one thinks all the +sentences. + +An "implior-indeterminate innperseq" is an innperseq in which one thinks +only each implicand and the outer segment it terminates. + +A "sententially indeterminate innperseq" is an innperseq in which one thinks +only the outer segment, and its inner endpoint, as it progresses +inward. +\end{sysrules} + + +Let us return to the matter of pure and descriptive syntax. The interest +of "Illusions" and "Innperseqs" is precisely that their abstract structure +cannot be separated from their physical and psychological character, and +thus that they are not isomorphic to any conventional ink-on-paper system. I +am trying to break through to unheard of, and hopefully significant, modes +of implication; to define implication structures (and derivation structures) +beyond the reach of past mathematics. + +\subsection{Constructed Memory Systems} + +In order to understand this section, it is necessary to be thoroughly +familiar with \essaytitle{Studies in Constructed Memories,} the essay following this +one. (I have not combined the two essays because their approaches are too +different.) I will define post-formalist systems in constructed memories, +beginning with a system in an M*-Memory. + +{ \centering \large "\textsc{Dream Amalgams}" \par} + +\begin{sysrules} +A "sentence" is a possible method, an $A_{a_i}$. with respect to an M*-Memory. +The sentence $A_{a_p}$ "implies" the sentence $A_{a_q}$ if and only if the $a_q$th +M*-assertion is actually thought; and either $A_{a_q} = A_{a_p}$, or else there is +cross-method contact of a mental state in $A_{a_q}$ with a state in $A_{q_p}$\footnote{sic?} + +The axioms must be chosen from sentences which satisfy two conditions. +The mental states in the sentences must have cross-method contact +with mental states in other sentences. And the M*-assertions +corresponding to the sentences must not be thought. + +Explanation: As \essaytitle{Studies in Constructed Memories} says, there can be +cross-method contact of states, because a normal dream can +combine totally different episodes in the dreamer's life into an +amalgam. +\end{sysrules} + +"\textsc{Dream Amalgams}" has Properties 1-5. For the first time, sentences are +structurally composite, with mental states being the relevant sentential +elements. Implication has an unusual character. The traditional type of +implication, modus ponens, is "directed," because the conditional is +directed. Even if $\ulcorner\varphi\supset\phi\urcorner$ is true +$\ulcorner\varphi\supset\phi\urcorner$ may not be. Now implication is also +directed in "\textsc{Dream Amalgams,}" but for a very different reason. +Cross-method contact, unlike the conditional, has a symmetric character. +What prevents implication from being necessarily symmetrical is that the +implicand's M*-assertion actually has to be thought, while the implior's +M*-assertion does not. Thus, implication is both subjective and mechanical, +it is subjective, in that it is a matter of volition which method is remembered +to have actually: been used. It is mechanical, in that when one is +remembering, one is automatically aware of the cross-method contacts of +states in $A_{a_q}$. The conditions on the axioms ensure that they will have +implications without losing Property 7. + +As for compound implication in "\textsc{Dream Amalgams,}" the organism +with the M*-Memory can't be aware of it at all; because it can't be aware +that at different times it remembered different methods to be the one +actually used. (In fact, the organism cannot be aware that the system has +Property 5, for the same reason.) On the other hand, to an outside observer +of the M*-Memory, indirect implication is not only thinkable but +mechanical. It is not superfluous because cross-method contact of mental +states is not necessarily transitive. The outside observer can decide whether a +sentence is a theorem by the following mechanical procedure. Check +whether the sentence's M*-assertion has acually been thought; if so, check all +sentences which imply it to see if any are axioms; if not, check all the +sentences which imply the sentences which imply it to see if any are axioms; +etc. The number of possible methods is given as finite, so the procedure is +certain to terminate. Again, an unprecedented mode of implication has been +defined. + +When a post-formalist system is defined in a constructed memory, the +discussion and analysis of the system become a consequence of constructed +memory theory and an extension of it. Constructed memory theory, which +is quite unusual but still more or less employs deductive inference, is used to +study post-formalist modes of inference which are anything but deductive. + +To aid in understanding the next system, which involves infalls in a +D-Memory, here is an + +{ \centering \large \framebox[1.1\width]{"Exercise to be Read Aloud"} \par} + +(Read according to a timer, reading the first word at O' O", and prolonging +and spacing words so that each sentence ends at the time in parentheses after +it. Do not pause netween sentences.) + +\begin{tabular}{ r l } + ($event_1$) & All men are mortal. (17") \\ + + ($Sentence_1=event_2s$) & The first utterance lasted 17" and ended at 17"; and lasted 15" and ended 1" ago. (59") \\ + + ($S_2=event_3$) & The second utterance lasted 42" and ended at 59": and lasted 50" and ended 2" ago. (1' 31") \\ + + ($S_3=event_4$) & The third utterance lasted 32" and ended at 1' 31"; and lasted 40" and ended 1" ago. (2' 16") \\ +\end{tabular} + +Since '32' in $S_3$ is greater than '2' in $S_2$, $S_2$ must say that $S_1$ ($=event_2$) +ended 30" after $S_2$ began, or something equally unclear. The duration of $S_2$ +is greater than the distance into the past to which it refers. This situation is +not a real infall, but it should give the reader some intuitive notion of an +infall. + + +\newcommand{\midheading}[1]{ + { \centering \large \textsc{#1} \par}} + +\midheading{"Infalls"} + +\begin{sysrules} + A "sentence" is a D-sentence, in a D-Memory such that $event_{j+1}$ is the first +thinking of the jth D-sentence, for all j. + +Two sentences "imply" another if and only if all three are the same; or else +the three are adjacent (and can be written $S_{j+1},S_j,S_{j-1}$), and are such +that $\delta_j=x_{j+1}-x_j> z_j,$ $S^D_{j-1}$ is the implicand. (The function of $S_{j+1}$ is to +give the duration $\delta_j=x_{j+1}-x_j$ of $S_j$. $S_j$ states that $event_j$, the first +thinking of $S^{D}_{j-1}$, ended at a distance $z_j$ into the past, where $z_j$ is smaller +than $S^D_j$'s own duration. The diagram indicates the relations.) +\end{sysrules} + +\img{infallsdiag} + +In this variety of D-Memory, the organism continuously thinks successive +D-sentences, which are all different, just as the reader of the above exercise +continuously reads successive and different sentences. Thus, the possibility +of repeating a sentence depends on the possibility of thinking it while one is +thinking another sentence---a possibility which may be far-fetched, but which +is not explicitly excluded by the definition of a "D-Memory." If the +possibility is granted, then "\textsc{Infalls}" has Properties 1--5. Direct implication is +completely mechanical; it is subjective only in that the involuntary +determination of the $z_j$ and other aspects of the memory is a 'subjective' +process of the organism. Compound implication is also mechanical to an +outside observer of the memory, but if the organism itself is to be aware of +it, it has to perform fantastic feats of multiple thinking. + +"\textsc{Dream Amalgams}" and "\textsc{Infalls}" are systems constructed with +imaginary elements, systems whose "notation" is drawn from an imaginary +object or system. Such systems have no descriptive syntax. Imaginary objects +were introduced into mathematics, or at least into geometry, by Nicholas +Lobachevski, and now I am using them as a notation. For these systems to +be nonisomorphic to any ink-on-paper systems, the mathematician must be +the organism with the M*-Memory or the D*-Memory. But this means that +in this case, the mathematics which is nonisomorphic to any ink-on-paper +system can be performed only in an imaginary mind. + +Now for a different approach. Carnap said that we are free to choose +the rules of a system arbitrarily. Let us take Carnap literally. I want to +construct more systems in constructed memories---so why not construct the +system by a procedure which ensures that constructed memories are +involved, but which is otherwise arbitrary? Why not suspend the striving +after "interesting" systems, that last vestige of the striving after +"correctness," and see what happens? Why not construct the rules of a +system by a chance procedure? + +To construct a system, we have to fill in the blanks in the following rule +schema in such a way that grammatically correct sentences result. + +\newcommand{\blankspace}{\_\_\_\_\_\_\_\_\_\_} + +\midheading{Rule Schema} + +\begin{sysrules} +A "sentence" is a(n) \blankspace. + +Two sentences "imply" a third if and only if the two sentences \blankspace\ the third. + +An "axiom" is a sentence that \blankspace. +\end{sysrules} + + +I now spread the pages of \essaytitle{Studies in Constructed Memories} on the floor. +With eyes closed, I hold a penny over them and drop it. I open my eyes and +copy down the expressions the penny covers. By repeating this routine, I +obtain a haphazard series of expressions concerning constructed memories. It +is with this series that I will fill in the blanks in the rule schema. In the next +stage, I fill the first (second, third) blank with the ceries of expressions +preceding the-first (second, third) period in the entire series. + +\midheading{"Haphazard System"} + +\begin{sysrules} +A "sentence" is a the duration D-sentences $\triangle\ (\mathparagraph^m)$ conclude these +"$\Phi*$-Reflection," or the future Assumption voluntarily force of +conviction for conclusion the Situation or by ongoing that this +system? be given telling between the Situation 1. + +Two sentences "imply" a third if and only if the two sentences is\slash was +contained not have to the acceptance that a certain and malleable +study what an event involves material specifically mathematics: +construct accompanies the rest, extra-linguistically image organism +can fantasy not remembering $\Phi*$-Memory, the future interval defined +in dream the third. + +An "axiom" is a sentence that internally D-sentences, just as the +"$\Phi*$-Memory" sentences $A_{a_1}$ is $A_{a_2}$. + +In the final stage, I cancel the smallest number of words I have to in +order to make the rules grammatical. +\end{sysrules} + +\midheading{"Fantasied Amnesia"} + +\begin{sysrules} +A "sentence" is a duration or the future force of conviction for the Situation +or this system given Situation 1. + +Two sentences "imply" a third if and only if the two sentences have the +acceptance that a certain and malleable study extra-linguistically can +fantasy not remembering the future interval defined in the third. + +An "axiom" is a sentence that internally just sentences $A_{a_2}$. +\end{sysrules} + +It becomes clear in thinking about "Fantasied Amnesia" that its +metametamathematics is dual. Describing the construction of the rules, the +metamathematics, by a systematic performance, is one thing. Taking the +finished metamathematics at face value, independently of its origin, and +studying it in the usual manner, is another. Let us take "Fantasied Amnesia" +at face value. As one becomes used to its rules, they become somewhat more +meaningful. I will say that an "interpretation" of a haphazard system is an +explanation of its rules that makes some sense out of what may seem +senseless. "Interpreting" is somewhat like finding the conditions for the +existence of a constructed memory which seemingly cannot exist. The first +rule of "Fantasied Amnesia" is a disjunction of three substantives. The +"Situation" referred to in the second substantive expression is either +Situation 1 or else an unspecified situation. The third substantive expression +apparently means "this system, assuming Situation 1," and refers to +"Fantasied Amnesia" itself. The definition of "sentence" is thus meaningful, +but very bizarre. The second rule speaks of "the acceptance" as if it were a +written assent. The rule then speaks of a "malleable study" as "fantasying" +something. This construction is quite weird, but let us try to accept it. The +third rule speaks of a sentence that "sentences" (in the legal sense) a possible +method. So much for the meaning of the rules. + +Turning to the nine properties of formalist systems, the reference to +"the future interval" in the implication rule of "Fantasied Amnesia" +indicates that the system has Property 2; and the system can perfectly well +have Property 8. It does not have Property 6 in any known sense. Certainly +it does have Property 9. it just might have Property. 1. But as for the other +four properties, it seems out of the question to decide whether "Fantasied +Amnesia" has them. For whatever it is worth, "Fantasied Amnesia" is on +balance incomparable to formalist systems. + +My transformation rule schema has the form of a biconditional, in +which the right clause is the operative one. If a transformation rule were to +vary, in such a way that it could be replaced by a constant rule whose right +clause was the disjunction of the various right clauses for the variable rule, +then the latter would vary "trivially." 1 will say that a system whose +transformation rule can vary non-trivially is a "heterodeterminate" system. +Since 1 have constructed a haphazard metamathematics, why not a +heterodeterminate metamathematics? Consider a mathematician with an +M-Memory, such that each $A_{a_i}$. is the consistent use of a different +transformation rule, a different definition of "imply," for the mathematics +in which the mathematician is discovering theorems. The consistent use of a +transformation rule is after all a method---a method for finding the +commitments premisses make, and for basing conclusions in premisses. When +the mathematician goes to remember which rule of inference he has actually +been using, he "chooses" which of the possible methods is remembered to +have actually been used. This situation amounts to a heterodeterminate +system. tn fact, the metamathematics cannot even be written out this time; I +can only describe it metametamathematically in terms of an imaginary +memory. + +We are now in the realm of mathematical systems which cannot be +written out, but can only be described metametamathematically. I will +present a final system of this sort. It is entitled \textsc{"System Such That No One +Knows What's Going On."} One just has to guess whether this system exists, +and if it does what it is like. The preceding remark is the +metametamathematical description, or definition, of the system. + +\subsection{Epilogue} + +Ever since Carnap's Principle of Tolerance opened the floodgates to +arbitrariness in mathematics, we have been faced with the prospect of a +mathematics which is indistinguishable from art-for-art's-sake, or +amusement-for-amusement's-sake. But there is one characteristic which saves +mathematics from this fate. Mathematics originated by abstraction from +primitive technology, and is indispensable to science and technology---in +short, mathematics has scientific applications. The experience of group +theory has proved, I hope once and for all, the bankruptcy of that narrow +practicality which would limit mathematics to what can currently be applied +in science. But now that mathematics is wide open, and anything goes, we +should be aware more than ever that scientific applicability is the only +objective value that mathematics has. I would not have set down constructed +memory theory and the post-formalist systems if I did not believe that they +could be applied. When and how they will be is another matter. + +And what about the "validity" of formalism? The rise of the formalist +position is certainly understandable. The formalists had a commendable, +rationalistic desire to eliminate the metaphysical problems associated with +mathematics. Moreover, formalism helped stimulate the development of the +logic needed in computer technology (and also to stimulate this paper). In +spite of the productiveness of the formalist position, however, it now seems +beyond dispute that formalism has failed to achieve its original goals. (My +pure philosophical writings are the last word on this issue.) Perhaps the main +lesson to be learned from the history of formalism is that an idea does not +have to be "true" to be productive. + + +\section*{Note} +Early versions of \textsc{"Illusions"} and \textsc{"Innperseqs"} appeared in my essay +"Concept Art," published in An Anthology, ed. La Monte Young, New +York, 1963. An early, July 1961 version of \textsc{"System Such That No One +Knows What's Going On"} appeared in dimension 14, Ann Arbor, 1963, +published by the University of Michigan College of Architecture and Design. + |