Changes

In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>

In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>

It is clear that some form of functional abstraction is being invoked in the above definition of <math>\operatorname{Nom}_1.\!</math>  Otherwise, the expression <math>x \mapsto {}^{\langle} x {}^{\rangle}\!</math> would indicate a constant function, one that maps every <math>x\!</math> in its domain to the same code or sign for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math>  But if this is allowed, then it seems either to invoke a more powerful concept, ''lambda abstraction'', than the one being defined or else to attempt an improper definition of the naming function in terms of itself.

It is clear that some form of functional abstraction is being invoked in the above definition of <math>\operatorname{Nom}_1.\!</math>  Otherwise, the expression <math>x \mapsto {}^{\langle} x {}^{\rangle}\!</math> would indicate a constant function, one that maps every <math>x\!</math> in its domain to the same code or sign for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math>  But if this is allowed, then it appears to pose a dilemma, either to invoke a more powerful concept of functional abstraction than the concept being defined, or else to attempt an improper definition of the naming function in terms of itself.

Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in reality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of higher order signs.

Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in actuality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of HO signs.

In CL contexts, especially in the Lisp tradition, the quotation operator is recognized as an "evaluation inhibitor" and implemented as a function that maps each syntactic element into its unique numerical identifier or "godel number".  Perhaps one should pause to marvel at the fact that a form of delay, deference, and interruption akin to an inhibition should be associated with the creation of signs that refer in meaningful ways.

In CL contexts, especially in the Lisp tradition, the quotation operator is recognized as an &ldquo;evaluation inhibitor&rdquo; and implemented as a function that maps each syntactic element into its unique numerical identifier or ''gödel number''.  Perhaps one should pause to marvel at the fact that a form of delay, deference, and interruption akin to an inhibition should be associated with the creation of signs that refer in meaningful ways.

On reflection, though, the connection between attribution and inhibition, or acknowledgment and deference, begins to appear less remarkable, and in time it can even be understood as natural and necessary.  For one thing, psychoanalytic and psychodynamic theories of mental functioning have long recognized that symbol formation and symptom formation are closely akin, being the twin founders of civilization and many of its discontents.  For another thing, the following etymology can be rather instructive:  The English word "memory" derives from the Latin "memor" for "mindful", which is akin to the Latin "mora" for "delay", the Greek "mermera" for "care", and the Sanskrit "smarati" for "he remembers".  To explore the verbal complex a bit further, it merits remembering that the ideas of "merit" and "membership", besides being connected with the due proportions, earned shares, and just deserts that are parceled out on parchment, are also tied up with the particular kind of care that is needed to take account of things part for part.  (The Latin "merere" for "earn" or "deserve", along with "membrana" for "skin" or "parchment" and "memor" for "mindful", are all akin to the Greek "merizein" for "divide" and "meros" for "part".)  Although the voices of psychology and etymology are seldom heard at this depth in the wilderness of formal abstraction, I think it is worth heeding them on this point.

On reflection, though, the connection between attribution and inhibition, or acknowledgment and deference, begins to appear less remarkable, and in time it can even be understood as natural and necessary.  For one thing, psychoanalytic and psychodynamic theories of mental functioning have long recognized that symbol formation and symptom formation are closely akin, being the twin founders of civilization and many of its discontents.  For another thing, the following etymology can be rather instructive:  The English word ''memory'' derives from the Latin ''memor'' for ''mindful'', which is akin to the Latin ''mora'' for ''delay'', the Greek ''mermera'' for ''care'', and the Sanskrit ''smarati'' for ''he remembers''.  To explore the verbal complex a bit further, it merits remembering that the ideas of ''merit'' and ''membership'', besides being connected with the due proportions, earned shares, and just deserts that are parceled out on parchment, are also tied up with the particular kind of care that is needed to take account of things part for part.  (The Latin ''merere'' for ''earn'' or ''deserve'', along with ''membrana'' for ''skin'' or ''parchment'' and ''memor'' for ''mindful'', are all akin to the Greek ''merizein'' for ''divide'' and ''meros'' for ''part''.)  Although the voices of psychology and etymology are seldom heard at this depth in the wilderness of formal abstraction, I think it is worth heeding them on this point.

In CL environments of the Pascal variety there are several different ways that HO signs can be created.  In these settings HO signs, or signs for referring to signs as objects, can be implemented as the "codes" that serve as numerical identifiers of characters or the "pointers" that serve as accessory indices of symbolic expressions.

In CL environments of the Pascal variety there are several different ways that higher order signs are created.  In these settings higher order signs, or signs for referring to signs as objects, can be implemented as the ''codes'' that serve as numerical identifiers of characters or the ''pointers'' that serve as accessory indices of symbolic expressions.

But not all the signs that are needed for referring to other signs can be constructed by means of quotation.  Other forms of HO signs have to be generated "de novo", that is, constructed independently of previous successions and introduced directly into their appropriate orders.  Among other things, this obviates the "obvious" strategy for telling the order of a sign by counting its quota of quotation marks.  Failing the chances of exploiting such a naive measure in absolute terms, and in the absence of a natural order for the construction of signs, the relative orders of signs can only be assessed by examining the complex network of denotative and connotative relationships that connect them, or the gaps that arise when they fail to do so.

But not all the signs that are needed for referring to other signs can be constructed by means of quotation.  Other forms of HO signs have to be generated "de novo", that is, constructed independently of previous successions and introduced directly into their appropriate orders.  Among other things, this obviates the "obvious" strategy for telling the order of a sign by counting its quota of quotation marks.  Failing the chances of exploiting such a naive measure in absolute terms, and in the absence of a natural order for the construction of signs, the relative orders of signs can only be assessed by examining the complex network of denotative and connotative relationships that connect them, or the gaps that arise when they fail to do so.