Directory:Jon Awbrey/Notes/Factorization And Reification - Revision history

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-The question is:  What sort of thing is a connotation?  Is it a sign?  That is to say, yet another term?  Or is it something like an abstract attribute, namely, a character, an intension, a property, or a quality?  And while we're asking, does it really even matter?
+The question is:  What sort of thing is a connotation?  Is it a sign?  &mdash; that is to say, yet another term?  Or is it something like an abstract attribute &mdash; a character, intension, property, or quality?  And while we're asking, does it really even matter?
 "No" is one answer worth considering.  But then:  Why does it not matter?  What reason could be given to excuse the indifference?

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-The question is:  What sort of thing is a connotation?  Is it a sign?  That is to say, is it yet another term?  Or is it something like an abstract attribute, namely, a character, an intension, a property, or a quality?  And while we're asking, does it really even matter?
+The question is:  What sort of thing is a connotation?  Is it a sign?  That is to say, yet another term?  Or is it something like an abstract attribute, namely, a character, an intension, a property, or a quality?  And while we're asking, does it really even matter?
-"No" is one answer worth considering.  But then:  Why does it not matter?  What reason might be given that would excuse the indifference?
+"No" is one answer worth considering.  But then:  Why does it not matter?  What reason could be given to excuse the indifference?
-This is a question that has exercised me since my earliest studies of Peirce.  I can remember discussing it with my philosophy mentor at the time and I distinctly recall having arrived at some conclusion or other, but, alas, I haven't the foggiest notion what exactly my revelation amounted to.  Perhaps that is all for the best, as the vagrancy of memory is frequently better than the vapidity of one's banalytic anamnesia.
+This is a question that has exercised me since my earliest studies of Peirce.  I can remember discussing it with my philosophy mentor at the time and I distinctly recall having arrived at some conclusion or other &mdash; alas, I haven't the foggiest notion what exactly my revelation amounted to.
 These days, I usually try to finesse the trick under the trumped up rubric of a factorization.  So let me excavate my last attempts to explain this business and see if I can improve on them.

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 What does all of this have to do with reification and so on?
-Well, suppose that the source domain <math>X\!</math> is a set of ''objects'', that the target domain <math>Y\!</math> is a set of ''signs'', and suppose that the function <math>f : X \to Y</math> indicates the effect of a classification, conceptualization, discrimination, perception, or some other type of sorting operation, distributing the elements of the set <math>X\!</math> of objects and into a set of sorting bins that are labeled with the elements of the set <math>Y,\!</math> regarded as a set of classifiers, concepts, descriptors, percepts, or just plain signs, whether these signs are regarded as being in the mind, as with concepts, or whether they happen to be inscribed more publicly in another medium.
+To begin answering that question, suppose that the source domain <math>X\!</math> is a set of ''objects'', that the target domain <math>Y\!</math> is a set of ''signs'', and suppose that the function <math>f : X \to Y</math> indicates the effect of a classification, conceptualization, discrimination, perception, or some other type of sorting operation, distributing the elements of the set <math>X\!</math> of objects and into a set of sorting bins that are labeled with the elements of the set <math>Y,\!</math> regarded as a set of classifiers, concepts, descriptors, percepts, or just plain signs, whether these signs are regarded as being in the mind, as with concepts, or whether they happen to be inscribed more publicly in another medium.
-In general, if we try to use the signs in the target codomain <math>Y\!</math> to reference the objects in the source domain <math>X,\!</math> then we will be invoking what used to be called &mdash; since the Middle Ages, I think &mdash; a manner of ''general reference'' or a mode of ''plural denotation'', that is to say, one sign will, in general, denote each of many objects, in a way that would normally be called ''ambiguous'' or ''equivocal''.
+In general, if we use the signs in the target domain <math>Y\!</math> to denote or describe the objects in the source domain <math>X,\!</math> then we are engaged in a form of ''general denotation'' or ''plural reference'' with regard to those objects, that is, each sign may refer to each of many objects, in a way that would normally be called ''ambiguous'' or ''equivocal''.
 Notice what I did not say here, that one sign denotes a ''set'' of objects, because I am for the moment conducting myself as such a dyed-in-the-wool nominal thinker that I hesitate even to admit so much as the existence of this thing we call a ''set'' into the graces of my formal ontology, though, of course, my casual speech is rife with the use of the word ''set'', and in a way that the nominal thinker, true-blue to the end, would probably be inclined or duty-bound to insist is a purely dispensable convenience.

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-The factorization of an arbitrary function into a surjective ("onto") function followed by an injective ("one-one") function is such a deceptively trivial observation that I had guessed that you would all wonder what in the heck, if anything, could possibly come of it.
+It may be difficult at first to see how anything of significance could follow from an observation so facile as the fact that an arbitrary function factors into a surjective function followed by an injective function.  What it means is that there is no loss of generality in assuming that there is a domain of intermediate entities under which the objects of a general denotation or plural reference can be marshaled, just as if they had something more essential and more substantial in common than the shared attachment to a coincidental name.  So the problematic status of a hypostatic entity like <math>o\!</math> is reduced from a question of its nominal existence to a matter of its local habitation.  Is it more like an object or more like a sign?  One wonders why there has to be only these two categories, and why not just form up another, but that does not seem like playing the game to propose it.  At any rate, I will defer for now one other obvious possibility &mdash; obvious from the standpoint of the pragmatic theory of signs &mdash; the option of assigning the new concept, or mental symbol, to the role of an interpretant sign.
 If we force the factored annotation function, initially extracted from the sign relation <math>L,\!</math> back into the frame from whence it came, we get the augmented sign relation <math>L^\prime,\!</math> shown in the next Figure:

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-But if you assimilate <math>i\!</math> to the realm of signs <math>S,\!</math> you will be showing your inclination to remain within the straight and narrow of ''conceptualist'' or even ''nominalist'' dogmas, and you may read this <math>i\!</math> as standing for an intelligible concept, or an ''idea'' of the safely decapitalized, mental impression variety.
+But if you assimilate <math>i\!</math> to the realm of signs <math>S,\!</math> you will be showing your inclination to remain within the straight and narrow of ''conceptualist'' or even ''nominalist'' dogmas, and you may regard the intermediate entity <math>i\!</math> as an intelligible concept, or an ''idea'' of the safely decapitalized, mental impression variety.
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-But if you dare to be truly liberal, you might just find that you can easily afford to accommmodate the illusions of both of these types of intellectual inclinations, and after a while you begin to wonder how all of that mental or ontological downsizing got started in the first place.
+But if you dare to be truly liberal, you might just find that you can easily afford to accommodate both intellectual inclinations, and after a while you begin to wonder how all that mental or ontological downsizing got started in the first place.
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-To sum up, we have recognized the perfectly innocuous utility of admitting the abstract intermediate object <math>i,\!</math> that may be interpreted as an intension, a property, or a quality that is held in common by all of the initial objects <math>o_j\!</math> that are plurally denoted by the sign <math>s.\!</math>  Further, it appears to be equally unexceptionable to allow the use of the sign <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to denote this shared intension <math>i.\!</math>  Finally, all of this flexibility arises from a universally available construction, a type of compositional factorization, common to the functional parts of the 2-adic components of any relation.
+To sum up, we have recognized the perfectly innocuous utility of admitting the abstract intermediate object <math>i_o,\!</math> that may be interpreted as an intension, a property, or a quality that is held in common by all of the initial objects <math>o_j\!</math> that are plurally denoted by the sign <math>s.\!</math>  Further, it appears to be equally unexceptionable to allow the use of the sign <math>{}^{\backprime\backprime} s \, {}^{\prime\prime}</math> to denote this shared intension <math>i_o.\!</math>  Finally, all of this flexibility arises from a universally available construction, a type of compositional factorization, common to the functional parts of the 2-adic components of any relation.
 ==Document History==

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 But if you dare to be truly liberal, you might just find that you can easily afford to accommmodate the illusions of both of these types of intellectual inclinations, and after a while you begin to wonder how all of that mental or ontological downsizing got started in the first place.
-{| align="center" cellspacing="10" style="text-align:center; width:90%"
+{| align="center" cellpadding="8" style="text-align:center"
+| [[Image:Factorization Sign Relation 5.jpg|500px]]
-<pre>
+|-
-o-----------------------------o
+| <math>\text{Figure 12.  Denotative Component of Sign Relation}~ L^{\prime\prime\prime}</math>
 |}

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 Let me now illustrate what I think that a lot of our controversies about nominalism versus realism actually boil down to in practice.  From a semiotic or a sign-theoretic point of view, it all begins with a case of ''plural reference'', which occurs when a sign <math>s\!</math> is taken to denote each object <math>o_j\!</math> in a collection of objects <math>\{ o_1, \ldots, o_k, \ldots \},</math> a situation whose general pattern is suggested by a sign-relational table of the following form:
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 For example, consider the sign relation <math>L\!</math> whose sign relational triples are precisely as shown in Table&nbsp;8.
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 Now, should you annex <math>i\!</math> to the object domain <math>O\!</math> you will have instantly given yourself away as having ''realist'' tendencies, and you might as well go ahead and call it an ''intension'' or even an ''Idea'' of the grossly subtlest Platonic brand, since you are about to booted from Ockham's Establishment, and you might as well have the comforts of your ideals in your exile.
-{| align="center" cellspacing="10" style="text-align:center; width:90%"
+{| align="center" cellpadding="8" style="text-align:center"
+| [[Image:Factorization Sign Relation 3.jpg|500px]]
-<pre>
+|-
-o-----------------------------o
+| <math>\text{Figure 10.  Denotative Component of Sign Relation}~ L^\prime</math>
 |}

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 The function <math>f : O \to S</math> factors into a composition <math>g \circ h,\!</math> where <math>g : O \to M,</math> and <math>h : M \to S,</math> as shown here:
-{| align="center" cellpadding="10" style="text-align:center; width:90%"
+{| align="center" cellpadding="8" style="text-align:center"
+| [[Image:Factorization Sign Relation Piece 2.jpg|500px]]
-<pre>
+|-
-o---------------------------------------o
+| <math>\text{Figure 5.  Factored Part of Sign Relation}~ L</math>
 |}