Changes

In many places these variegated styles of usage express themselves not so much in isolated domains of influence or distinctive layers of context as in different perspectives on the same text.  But different lights on a developing picture can cause different figures and patterns to emerge, and different ways of treating a developing text can lead it to grow in different directions.  Thus, discrepant points of view on the emergence of a literature can stimulate different works to vie for its canon, and discriminating angles of approach to what seems like a level plain and a unified field of language can harvest a wealth of alternate appreciations.  And so different styles of writing arise in correspondence with different styles of reading, and each rising style of readership engenders a new style of authorship in its wake.

In many places these variegated styles of usage express themselves not so much in isolated domains of influence or distinctive layers of context as in different perspectives on the same text.  But different lights on a developing picture can cause different figures and patterns to emerge, and different ways of treating a developing text can lead it to grow in different directions.  Thus, discrepant points of view on the emergence of a literature can stimulate different works to vie for its canon, and discriminating angles of approach to what seems like a level plain and a unified field of language can harvest a wealth of alternate appreciations.  And so different styles of writing arise in correspondence with different styles of reading, and each rising style of readership engenders a new style of authorship in its wake.

At other times these degrees of formality play themselves out in a temporal process.  Consider a typical scenario for solving problems through formalization:

At other times these degrees of formality play themselves out in a temporal process.  Consider a typical scenario for solving problems through formalization:

1. One begins by approaching the problem informally, in other words, in IL posed terms, drawing on the common resources of technical notions and mathematical methods that are available, familiar, intuitively understood, and that suggest themselves as possibly being relevant to the problem.

# One begins by approaching the problem informally, in other words, in IL-posed terms, drawing on the common resources of technical notions and mathematical methods that are available, familiar, intuitively understood, and that suggest themselves as possibly being relevant to the problem.

 

# Next, the problem of interest and the array of methods selected for addressing it are both reformulated in FL terms, a process that requires many obscurities and omissions of the original problem statement to be weeded out and filled in, respectively.

2. Next, the problem of interest and the array of methods selected for addressing it are both reformulated in FL terms, a process that requires many obscurities and omissions of the original problem statement to be weeded out and filled in, respectively.

# Finally, the formalized version of the problem method constellation is reconstructed to the extent possible in a CL framework.

 

3. Finally, the formalized version of the problem method constellation is reconstructed to the extent possible in a CL framework.

At any stage of this procedure one may discover, or begin to suspect, that the current representation of the problem or the present selection of methods is inadequate to the task or unlikely to lead to a solution.  In this event one is forced to back track to an earlier stage of the problem's formulation and to look for ways of changing one's grasp of the situation.

At any stage of this procedure one may discover, or begin to suspect, that the current representation of the problem or the present selection of methods is inadequate to the task or unlikely to lead to a solution.  In this event one is forced to back track to an earlier stage of the problem's formulation and to look for ways of changing one's grasp of the situation.

Even though the styles of usage at the three degrees of formalization use overlapping vocabularies of technical terms, the interpretations that they put on some of these terms, together with the working attitudes that they promote toward the corresponding concepts, are tantamount in practice to the possession of distinct concepts for the very same terms.

Even though the styles of usage at the three degrees of formalization use overlapping vocabularies of technical terms, the interpretations that they put on some of these terms, together with the working attitudes that they promote toward the corresponding concepts, are tantamount in practice to the possession of distinct concepts for the very same terms.

Three issues of linguistic usage on which these three NOSs get most out of joint are on the questions of (1) signs and their significance, (2) the utilization of set theory and set theoretic constructions, and (3) the ontological or pragmatic status of variables.  The rest of this section makes a cursory survey of the bearings that the three NOSs take toward these issues, in preparation for more detailed treatments in later sections.

Three issues of linguistic usage on which the three norms of significance get most out of joint are on the questions of (1) signs and their significance, (2) the utilization of set theory and set-theoretic constructions, and (3) the ontological or pragmatic status of variables.  The rest of this section makes a cursory survey of the bearings that the three norms take toward these issues, in preparation for more detailed treatments in later sections.

In each perspective that an observer takes up, the natural attitude is to focus on a particular class of objects, to remain less aware of the signs being used to denote them, and to remain even less aware that these objects and signs can take up other roles in the same or other sign relations.  In constantly shifting from one perspective to another, however, the transparent uses of signs and the ulterior circumstances that determine how objects and signs are cast start to become visible.  Altogether, the interaction between casual and formal styles of usage is like an exchange carried on between radically different economies, where commodities and utilities that are freely traded in one kind of market are severely taxed in the other.

In each perspective that an observer takes up, the natural attitude is to focus on a particular class of objects, to remain less aware of the signs being used to denote them, and to remain even less aware that these objects and signs can take up other roles in the same or other sign relations.  In constantly shifting from one perspective to another, however, the transparent uses of signs and the ulterior circumstances that determine how objects and signs are cast start to become visible.  Altogether, the interaction between casual and formal styles of usage is like an exchange carried on between radically different economies, where commodities and utilities that are freely traded in one kind of market are severely taxed in the other.

The IL perspective, along with its specialization to ordinary mathematical discourse, thinks itself to have a grasp of the unitary object itself, conveniently forgetting the multiplicity of abstract, arbitrary, and artificial constructions that are needed to make this impression possible.  In particular, the ordinary mathematical attitude thinks itself to have a grasp of the one idea while its puts the many appearances out of mind, and it constantly exerts itself to neglect all the labor that goes into taking up this stance.  It ignores the circumstance that numbers, however intuited, can only be indicated and rationalized to others as equivalence classes of constructions formed on the matter of numerals.

The IL perspective, along with its specialization to ordinary mathematical discourse, thinks itself to have a grasp of the unitary object itself, conveniently forgetting the multiplicity of abstract, arbitrary, and artificial constructions that are needed to make this impression possible.  In particular, the ordinary mathematical attitude thinks itself to have a grasp of the one idea while its puts the many appearances out of mind, and it constantly exerts itself to neglect all the labor that goes into taking up this stance.  It ignores the circumstance that numbers, however intuited, can only be indicated and rationalized to others as equivalence classes of constructions formed on the matter of numerals.

The FL perspective, along with its implementations in CL contexts, allows one to treat signs as objects, and thus to study syntactic domains as objective languages.  This creates what seems like a higher order of discussion, but the designation of these objects as "signs" is purely token if their use as signs is forgotten in the process.  Consequently, the FL perspective, together with the CL attitude that realizes it, has the job of recovering and reconstructing exactly what has been taken out of consideration in the IL context:  those details of actual usage that are taken for granted, abstracted away, and conveniently ignored.

The FL perspective, along with its implementations in CL contexts, allows one to treat signs as objects, and thus to study syntactic domains as objective languages.  This creates what seems like a higher order of discussion, but the designation of these objects as ''signs'' is purely token if their use as signs is forgotten in the process.  Consequently, the FL perspective, together with the CL attitude that realizes it, has the job of recovering and reconstructing exactly what has been taken out of consideration in the IL context, namely, the details of actual usage that are taken for granted, abstracted away, and conveniently ignored.

Although the mathematical structures developed under the informal NOS can become incredibly sophisticated in their orders of complexity and degrees of formalization, from a pragmatic standpoint they are still construed under naive assumptions about language use.  This is because discussions carried out under the IL perspective do not make it their business to reflect on the relations between objects and signs, but presume that these matters can be separated from their subject proper and relegated to preliminary stages of the ultimately refined treatment.

Although the mathematical structures developed under the informal norm of significance can become incredibly sophisticated in their orders of complexity and degrees of formalization, from a pragmatic standpoint they are still construed under naive assumptions about language use.  This is because discussions carried out under the IL perspective do not make it their business to reflect on the relations between objects and signs, but presume that these matters can be separated from their subject proper and relegated to preliminary stages of the ultimately refined treatment.

In order to make this discussion of styles and issues and more concrete, the next several sections examine the practical bearings of the three styles of usage as they work out with regard to each of the identified issues of usage.  This will be done by choosing a theoretical subject to illustrate the ideals of each style of usage, and then by developing the bearing of this subject on each of the three issues mentioned.

In order to make this discussion of styles and issues and more concrete, the next several sections examine the practical bearings of the three styles of usage as they work out with regard to each of the identified issues of usage.  This will be done by choosing a theoretical subject to illustrate the ideals of each style of usage, and then by developing the bearing of this subject on each of the three issues mentioned.

In accord with this plan, the next three sections present the basic ideas of three subjects:  group theory, formal language theory, and computation theory.  The presentation of these subjects is intended to serve both illustrative and instrumental purposes, exemplifying the ideals of the IL, FL, and CL styles of usage, respectively, but also equipping subsequent discussion with a supply of ready tools that can be used in its further development.  After the treatment of these three subjects, and following the introduction of higher order sign relations, the next three sections after that are finally able to take up the three issues mentioned above, concerning the theoretical standings of signs, sets, and variables, respectively, and to consider how each of these issues appears in the light of each style of usage.

In accord with this plan, the next three sections present the basic ideas of three subjects:  group theory, formal language theory, and computation theory.  The presentation of these subjects is intended to serve both illustrative and instrumental purposes, exemplifying the ideals of the IL, FL, and CL styles of usage, respectively, but also equipping subsequent discussion with a supply of ready tools that can be used in its further development.  After the treatment of these three subjects, and following the introduction of higher order sign relations, the next three sections after that are finally able to take up the three issues mentioned above, concerning the theoretical standings of signs, sets, and variables, respectively, and to consider how each of these issues appears in the light of each style of usage.

===6.6. Basic Notions of Group Theory===

===6.6. Basic Notions of Group Theory===