12,080
edits
Changes
Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 11:01, 16 August 2011
, 11:01, 16 August 2011→6.28. The Bridge : From Obstruction to Opportunity: + raw text
===6.28. The Bridge : From Obstruction to Opportunity===
===6.28. The Bridge : From Obstruction to Opportunity===
<pre>
There are many reasons for using IR's to describe formal objects, especially as the size and complexity of these objects grows beyond the bounds of finite information capacities to represent in practical terms. This is extremely pertinent to the progress of the present discussion. As often happens, when a top down investigation of complex families of formal objects actually succeeds in arriving at examples that are simple enough to contemplate in extensional terms, it can be difficult to see the relation of such impoverished examples to the cases of original interest, all of them typically having infinite cardinality and indefinite complexity. In short, once a discussion is brought down to the level of its smallest cases it can be nearly impossible to bring it back up to the level of its intended application. Without invoking IR's of sign relations there is little hope that this discussion can rise far beyond its present level, eternally elaborating the subtleties of cases as elementary as A and B.
There are many obstacles to building this bridge, but if these forms of obstruction are understood in the proper fashion, it is possible to use them as stepping stones, to capitalize on their redoubtable structures, and to convert their recalcitrant materials into a formal calculus that can serve the aims and means of instruction.
This approach requires me to consider a chain of relationships that connects signs, names, concepts, properties, sets, and objects, along with various ways that these classes of entities have been viewed at different periods in the development of mathematical logic.
I would like to begin by giving an "impressionistic capsule history" of the relevant developments in mathematical logic, admittedly as viewed from a certain perspective, but hoping to allow room for alternative persepectives to have their way/ present themselves in their own best light.
The human mind, boggling at the many to many relation between objects and signs that it finds in the world as soon as it begins to reflect on its own reasoning process, hits upon the strategy of interposing a realm of intermediate nodes between objects and signs, and looking through this medium for ways to factor the original relation into simpler components.
At the beginning of logic, the human mind, as soon as it begins to reflect on its own reasoning process, boggles at the many to many relation between objects and signs that it finds itself conducting through the world.
There are two methods for attempting to disentangle this confusion that are generally tried, the first more rarely, the second quite frequently, though apparently in opposite proportion to their respective chances of actual success. In order to describe the rationales of these methods I need to introduce a number of technical concepts.
Suppose that P and Q are dyadic relations, with P c XxY and Q c YxZ. Then the "contension" of P and Q is a triadic relation R c XxYxZ that is notated as R = P&Q and defined as follows:
P&Q = {<x, y, z> C XxYxZ : <x, y> C P and <y, z> C Q}.
In other words, P&Q is the intersection of the "inverse projections" P' = Pr12 1(P) and Q' = Pr23 1(Q), which are defined as follows:
Pr12 1(P) = PxZ = {<x, y, z> C XxYxZ : <x, y> C P}.
Pr23 1(Q) = XxQ = {<x, y, z> C XxYxZ : <y, z> C Q}.
Inverse projections are often referred to as "extensions", in spite of the conflict this creates with the "extensions" of terms, concepts, and sets.
One of the standard turns of phrase that finds use in this setting, not only for translating between ER's and IR's, but for converting both into computational forms, is to associate any set S contained in a space X with two other types of formal objects: (1) a logical proposition pS known as the characteristic, indicative, or selective proposition of S, and (2) a binary valued function fS: X >B known as the characteristic, indicative, or selective function of S.
Strictly speaking, the logical entity pS is the IR of the tribe, presiding at the highest level of abstraction, while fS and S are its concrete ER's, rendering its concept in functional and geometric materials, respectively. Whenever it is possible to do so without confusion, I try to use identical or similar names for the corresponding objects and species of each type, and I generally ignore the distinctions that otherwise set them apart. For instance, in moving toward computational settings, fS makes the best computational proxy for pS, so I commonly refer to the mapping fS: X >B as a "proposition" on X.
Regarded as logical models, the elements of the contension P&Q satisfy the proposition referred to as the "conjunction of extensions" P' and Q'.
Next, the "composition" of P and Q is a dyadic relation R' c XxZ that is notated as R' = P.Q and defined as follows:
P.Q = Pr13(P&Q) = {<x, z> C XxZ : <x, y, z> C P&Q}.
In other words:
P.Q = {<x, z> C XxZ : <x, y> C P and <y, z> C Q}.
Using these notions, the customary methods for disentangling a many to many relation can be explained as follows:
1.
2.
In the logic of the ancients, the many to one relation of things to general names ...
In early approaches to mathematical logic, from Leibniz to Peirce and Frege, one ordinarily spoke of the extensions and intensions of concepts.
Typically, one starts a work of bridge building by casting a thin line across the intervening gap, using this expediency to conduct a slightly more substantial linkage over the rift, and then proceeding through a train of successors to draw increasingly stronger connections between the opposing shores until a load bearing framework can be established. There is an analogue of this operation that fits the current situation, and this is something I can do this by taking up the sign relations A and B, already introduced in extensional terms, and redescribing the abstract features of their structures in intensional terms.
This would be the ideal plan. But bridging the "tensions", ex and in , that subsist within the forms of representation is not as easy as that. In order to convey the importance of the task and provide a motivation for carrying it out, I will plot a chain of relationships that stretches from signs, names, and concepts to properties, sets, and objects.
As a way of resolving the discriminated "tensions", posed here to fall into "ex " and "in " kinds, the strategy just described affords a way of approaching the problem that is less like a bridge than a pole vault, taking its pivot on a fixed set of narrowly circumscribed sign relations to make a transit from extensional to intensional outlooks on their form. With time and reflection, the logical depth of the supposed distinction, the "pretension" of maintaining a couple of separate but equal tensions in isolation from each other, does not withstand a persistent probing. Accordingly, the gulf between the two realms can always be fathomed by a finitely informed creature, in fact, by the very form of interpreter that created the fault in the first place. Consequently, converting the form of a transient vault into the substance of a usable bridge requires in adjunction only that initially pliable and ultimately tensile sorts of connecting lines be conducted along the tracery of the vault until the work of castling the gap can begin.
In the pragmatic theory of signs, the word "representation" is a technical term that is synonymous with the word "sign", in other words, it applies to an entity in the most general category of things that can enter into sign relations in the roles of signs and interpretants. Thus, in this usage the scope of the term "representation" includes all sorts of syntactic, descriptive, and conceptual entities, a range of options I will frequently find it convenient to suggest by drawing on a pair of stock phrases: "terms and concepts" (TAC's) in a conjuctive context, versus "terms or concepts" (TOC's) in a disjunctive context.
In mathematics, the word "representation" is commonly reserved for referring to a "homomorphism", that is, a linear transformation or a structure preserving mapping h: X >Y between mathematical "objects", that is, structure bearing spaces in a category of comparable domains.
In keeping with the spirit of the current discussion, I will first present a set of examples that are designed to illustrate what I mean by an IR. In general, an IR of any object is a sign, description, or concept that denotes, describes, or conceives its object in terms of its properties, that is, in terms of the logical attributes that the object possesses or the propositional features that the object is supposed to have. If the object to be represented is a complex formal object like a sign relation, then there needs to be an IR of each elementary sign relation and an IR of the sign relation as a whole.
But first, before I try to tackle this project, it is advisable to seek a measure of theoretical advantage that I can bring to bear on the task. This I can do by anchoring my focal outlook on sign relations within a more global consideration of n place relations. Not only will this help with the conceptual recasting of A and B, but it will also support later stages of the present work, especially in the effort to build a collection of readily accessible linkages between the extensions and the intensions of each construct that I try to use, and ultimately of each concept and term that might conceivably find a use in inquiry.
</pre>
===6.29. Projects of Representation===
===6.29. Projects of Representation===