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diff --git a/essays/post_formalism_memories.tex b/essays/post_formalism_memories.tex index 73477a5..0c867a2 100644 --- a/essays/post_formalism_memories.tex +++ b/essays/post_formalism_memories.tex @@ -3,8 +3,12 @@ \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 @@ -22,14 +26,18 @@ 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 +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 -trasformation rule is the rule that $\psi$ may be concluded from $\varphi$ and +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},} @@ -406,6 +414,7 @@ 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 @@ -669,7 +678,8 @@ 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. -\subsection{Epilogue} +\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 |