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Diffstat (limited to 'essays/post_formalism_memories.tex')
| -rw-r--r-- | essays/post_formalism_memories.tex | 55 |
1 files changed, 26 insertions, 29 deletions
diff --git a/essays/post_formalism_memories.tex b/essays/post_formalism_memories.tex index bf7decc..9b77855 100644 --- a/essays/post_formalism_memories.tex +++ b/essays/post_formalism_memories.tex @@ -1,3 +1,6 @@ +\newcommand{\midheading}[1]{ + { \vskip 1em \centering \large \textsc{#1} \par \vskip 1em }} + \chapter{Post-Formalism in Constructed Memories} \section{Post-Formalist Mathematics} @@ -125,7 +128,7 @@ Compound indirect implication is a puzzle. Now for the first post-formalist system. -{ \centering \large "\textsc{Illusions}" \par} +\midheading{"Illusions"} \begin{sysrules} A "sentence" is the following page (with the figure on it) so long as the @@ -151,7 +154,7 @@ figures with a variety of real (measured) ratios and practicing judging these ratios; and so forth. \end{sysrules} -\img{illusions} +\imgw{2in}{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 @@ -159,7 +162,7 @@ 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} +\imgw{4in}{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 @@ -202,7 +205,7 @@ 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} +\midheading{"Order"} \begin{sysrules} A "sentence" is a member of a finite set of integers. @@ -218,7 +221,7 @@ 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} +\imgw{4in}{img/orderstructure} where loops indicate multiple occurances of the same sentence, could adequately represent a proof in "Order," but could not remotely represent @@ -229,7 +232,7 @@ onto the paper. This is utterly impossible. Here is the next system. -{ \centering \large "\textsc{Innperseqs}" \par} +\midheading{"Innperseqs"} \begin{sysrules} Explanation: Consider the rainbow halo which appears to surround a small @@ -261,20 +264,12 @@ 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} +{\centering +\begin{minipage}{1.6in}\imgw{1.3in}{img/innperseqs}\end{minipage} + \begin{minipage}{2.25in} "Sentences" at \begin{tabular}{ c r l } @@ -318,7 +313,12 @@ $(d,b,a_6)$ $(e,c,a_7)$ $(f,e,d)$ $(g)$ + \end{minipage}\par} +In the diagram, 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. 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 @@ -354,7 +354,8 @@ 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} +\midheading{Indeterminacy} + \begin{sysrules} A "totally determinate innperseq" is an innperseq in which one thinks all the sentences. @@ -384,7 +385,7 @@ 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} +\midheading{"Dream Amalgams"} \begin{sysrules} A "sentence" is a possible method, an $A_{a_i}$. with respect to an M*-Memory. @@ -445,13 +446,13 @@ 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} +{ \vskip 1.5em \centering \large \framebox[1.1\width]{"Exercise to be Read Aloud"} \par\vskip 1.5em} (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 } +\begin{tabular}{ r p{2.5in} } ($event_1$) & 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") \\ @@ -467,10 +468,6 @@ 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} @@ -485,7 +482,7 @@ thinking of $S^{D}_{j-1}$, ended at a distance $z_j$ into the past, where $z_j$ than $S^D_j$'s own duration. The diagram indicates the relations.) \end{sysrules} -\img{infallsdiag} +\imgw{4in}{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 @@ -547,7 +544,7 @@ preceding the-first (second, third) period in the entire series. \begin{sysrules} A "sentence" is a the duration D-sentences $\triangle\ (\mathparagraph^m)$ conclude these -"$\Phi*$-Reflection," or the future Assumption voluntarily force of +"$\Phi^*$-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. @@ -555,11 +552,11 @@ 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 +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 $A_{a_2}$. +"$\Phi^*$-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. |