change usage of math mode

1 files changed, 18 insertions, 18 deletions

diff --git a/essays/post_formalism_memories.tex b/essays/post_formalism_memories.tex
index 6f7d45b..6cf9b4d 100644
--- a/essays/post_formalism_memories.tex
+++ b/essays/post_formalism_memories.tex
@@ -185,7 +185,7 @@ 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 
+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 
@@ -234,8 +234,8 @@ the following trivial formalist system.
 \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. 
+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}
@@ -412,19 +412,19 @@ 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. 
+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. 
+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 
+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 
+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 
@@ -442,8 +442,8 @@ $\ulcorner\phi\supset\varphi\urcorner$ 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, 
+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 
@@ -451,15 +451,15 @@ 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 
+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 
+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 
+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 
@@ -503,8 +503,8 @@ infall.
 \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 $j$. 
+	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 
@@ -522,12 +522,12 @@ thinking of $S^{D}_{j-1}$, ended at a distance $z_j$ into the past, where $z_j$
 	\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 
+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 
+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}
@@ -541,7 +541,7 @@ 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 
+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.