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===6.13. Issue 2. The Status of Sets===
===6.13. Issue 2. The Status of Sets===
<pre>
That the word "set" is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new. (Julius Konig, 1905).
Set theory is not as young as it used to be, and not half as naive as it was when this statement was originally made, but the statement itself is just as apt in its application to the present scene and just as fresh in its lack of novelty as it was then. In the current setting, though, I am not so concerned with potentially different theoretical notions of a set that are represented by conventionally different axiom systems as I am with the actual diversity of practical notions that are used to deal with sets under each of the three NOS's identified.
Even though all three NOS's use set theoretic constructions, the implicit theories of sets that are involved in their different uses are so varied in their assumptions and intentions that it amounts to a major source of friction between the casual and formal styles to try to pretend that the same subject is being invoked in every case. In particular, it makes a huge difference whether these sets are treated objectively, as belonging to the OF, or treated syntactically, as belonging to the IF.
In practical terms it makes all the difference in the world whether a set is viewed as a set of objects or whether it is viewed as a set of signs. The same set can be contemplated in each type of placement, but it does not always fit as well into both types of role. A set of objects is properly a part of the OF, and this is intended in its typical parts to model those realities whose laws and vagaries can extend outside the means of an agent's control. A set of signs is properly part of the IF, and this is constructed in its typical parts so that its variations and selections are subject to control for the ends of interpretive indication. The relevant variable is one of control, and the measure of it tells how well matched are the proper placements and the typical assignments that a given set is given.
Things referred to the objective world are not things that one expects to have much control over, at least, not at first, even though a reason for developing a language is to gain more control over events in time. Things referred to the realm of signs are things that one thinks oneself to have under control, at least, at first, even though their complexity can evolve in time beyond one's powers of oversight.
In an ordinary mathematical context, when one writes out the expression for a finite set in the form "{x1, ..., xn}", then one expects to see the names of objects appearing between the braces. Furthermore, even if these additional expectations are hardly ever formalized, these objects are typically expected to be the terminal objects of denotative value in the appropriate context of discussion and to inhabit a single order of objective existence. In other words, it is common to assume that all of the objects named have the same type, with no relations of consequence, functional, semantic, or otherwise, obtaining among them. As soon as these assumptions are made explicit, of course, it is obvious that they do not have to be so.
In FL contexts, when a set is taken as the alphabet or the lexicon of a formal language, then the objects named are themselves signs, but it is still only their names that are subject to appearing between the braces. Often one seeks to handle this case by saying that what really appears between the braces are signs of sort that can suffice to represent themselves, and thus that these signs literally constitute their own names, but this is not ultimately a sensible tactic to try. As always, only the tokens of signs can appear on the page, and these come and go as the pages are turned. Although these tokens, by representing the types that encase them, partly succeed in referring to themselves, what they denote on principle is something much more abstract, general, and invariant than their own concrete, particular, and transient selves. Nevertheless, the expectation that all of the elements in the set reside at the same level of syntactic existence is still in effect.
The construction of a RIF demands a closer examination of these assumptions and requires a single discussion that can refer to mixed types of elements with significant relations among them.
In a FL context one needs to be more self conscious about the use of signs, and, after an initially painful period during which critical reflection seems more to interfere with thought more than to facilitate understanding, it is hoped that the extra measure of reflection will pay off when it is time to mediate one's thinking in a CL framework.
There are numerous devices that one can use to assist with the task of reflection. Rather than trying to divert the customary connections of informal language use and the conventional conduct of its interpretation, it is easier to introduce a collection of markedly novel signs, analogous to those already in use but whose interpretation is both free enough to be changed and controlled through a series of experimental variations and flexible enough to be altered when fitting and repaired when faulty.
If X = {x1, ... , xn} is a set of objects under discussion, then one needs to be able to consider several sets of signs that might be associated, element by element, with the elements of X.
1. The "nominal resource" ("nominal alphabet" or "nominal lexicon") for X is a set of signs that is notated and defined as follows:
X" = Nom (X) = {"x1", ... , "xn"}.
This concept is intended to capture the ordinary usage of this set of signs in one familiar context or another.
2. The "mediate resource" ("mediate alphabet" or "mediate lexicon") for X is a set of signs that is notated and defined as follows:
X<> = Med (X) = {<x1>, ... , <xn>}.
This concept provides a middle ground between the nominal resource above and the literal resource below.
3. The "literal resource" ("literal alphabet" or "literal lexicon")
for X is a set of signs that is notated and defined as follows:
X = Lit (X) = { x1 , ... , xn }.
This concept is intended to supply a set of signs that can be used in ways that are analogous to familiar usages, but which are more subject to free variation and thematic control.
</pre>
===6.14. Issue 3. The Status of Variables===
===6.14. Issue 3. The Status of Variables===
