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Directory:Jon Awbrey/Notes/Factorization Issues (view source)

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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.
 
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.
   −
<pre>
+
What it means is that &mdash; without loss or gain of generality &mdash; we might as well assume that there is a domain of intermediate entities under which the objects of a general denotation can be marshalled, just as if they actually had something rather more essential and really more substantial in common than the shared attachment to a coincidental name. So the problematic status of a hypostatic entity like <math>x\!</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.
−
What it means is that, "without loss or gain of generality" (WOLOGOG),
  −
we might as well assume that there is a domain of intermediate entities
  −
under which the objects of a general denotation can be marshalled, just
  −
as if they actually had something rather more essential and really more
  −
substantial in common than the shared attachment to a coincidental name.
  −
So the problematic status of a hypostatic entity like x is reduced from
  −
a question of its nominal existence to a matter of its local habitation.
  −
Is it very like a sign, or is it rather more like an object?  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 --
  −
obvious from the standpoint of the pragmatic theory of signs --
  −
the option of assigning the new concept, or mental symbol,
  −
to the role of an interpretant sign.
     −
If we force the factored annotation function,
+
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:
−
initially extracted from the sign relation L,
  −
back into the frame from whence it once came,
  −
we get the augmented sign relation L', shown
  −
in the next vignette:
  −
</pre>
      
{| align="center" cellpadding="10" style="text-align:center; width:90%"
 
{| align="center" cellpadding="10" style="text-align:center; width:90%"
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|}
 
|}
   −
<pre>
+
This amounts to the creation of a hypostatic object <math>x,\!</math> which affords us a singular denotation for the sign <math>y.\!</math>
−
This amounts to the creation of a hypostatic object x,
  −
which affords us a singular denotation for the sign y.
     −
By way of terminology, it would be convenient to have
+
By way of terminology, it would be convenient to have a general name for the transformation that converts a bare, ''nominal'' sign relation like <math>L\!</math> into a new, improved ''hypostatically augmented or extended'' sign relation like <math>L^\prime.</math>  Let us call this kind of transformation an ''objective extension'' or an ''outward extension'' of the underlying sign relation.
−
a general name for the transformation that converts
  −
a bare "nominal" sign relation like L into a new,
  −
improved "hypostatically augmented or extended"
  −
sign relation like L'.
     −
I call this kind of transformation
+
This naturally raises the question of whether there is also an augmentation of sign relations that might be called an ''interpretive extension'' or an ''inward extension'' of the underlying sign relation, and this is the topic that I will take up next.
−
an "objective extension" (OE) or
  −
an "outward extension" (OE) of
  −
the underlying sign relation.
     −
This naturally raises the question of
+
==Nominalism and Realism==
−
whether there is also an augmentation
  −
of sign relations that might be called
  −
an "interpretive extension" (IE) or
  −
an "inward extension" (IE) of
  −
the underlying sign relation,
  −
and this is the topic that
  −
I will take up next.
  −
</pre>
     −
==Nominalism and Realism==
+
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 happens when a sign <math>y\!</math> is quite literally taken to denote each object <math>x_j\!</math> in a whole collection of objects <math>\{ x_1, \ldots, x_k, \ldots \},</math> a situation that can be represented in a sign-relational table like this one:
    +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 
<pre>
 
<pre>
−
Let me illustrate what I think that a plethora 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 happens when a sign y
  −
is quite literally taken to denote each object x_j in a whole
  −
collection of objects {x_1, ..., x_k, ...}, a situation that
  −
I'd normally represent in a sign-relational table like so:
  −
   
o---------o---------o---------o
 
o---------o---------o---------o
 
| Object  |  Sign  | Interp  |
 
| Object  |  Sign  | Interp  |
Line 270: Line 228:  
|  ...  |    y    |  ...  |
 
|  ...  |    y    |  ...  |
 
o---------o---------o---------o
 
o---------o---------o---------o
 +
</pre>
 +
|}
   −
For brevity, let us consider the sign relation L
+
For brevity, let us consider a sign relation <math>L\!</math> whose relational database table is precisely this:
−
whose relational database table is precisely this:
      +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
|      Sign Relation L      |
 
|      Sign Relation L      |
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|  x_3  |    y    |  ...  |
 
|  x_3  |    y    |  ...  |
 
o---------o---------o---------o
 
o---------o---------o---------o
 +
</pre>
 +
|}
    
For the moment, it does not matter what the interpretants are.
 
For the moment, it does not matter what the interpretants are.
   −
I would like to diagram this somewhat after the following fashion,
+
I would like to diagram this somewhat after the following fashion, here detailing just the denotative component of the sign relation, that is, the 2-adic relation that is obtained by "projecting out" the Object and Sign columns of the table.
−
here detailing just the denotative component of the sign relation,
  −
that is, the 2-adic relation that is obtained by "projecting out"
  −
the Object and the Sign columns of the table.
      +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L  |
 
| Denotative Component of L  |
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|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
   −
I would like to -- but my personal limitations in the
+
I would like to &mdash; but my personal limitations in the Art of ASCII Hieroglyphics do not permit me to maintain this level of detail as the figures begin to ramify much beyond this level of complexity.  Therefore, let me use the following device to symbolize the same configuration:
−
Art of ASCII Hieroglyphics do not permit me to maintain
  −
this level of detail as the figures begin to ramify much
  −
beyond this level of complexity.  Therefore, let me use
  −
the following device to symbolize the same configuration:
      +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L  |
 
| Denotative Component of L  |
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|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
    
Notice the subtle distinction between these two cases:
 
Notice the subtle distinction between these two cases:
   −
  1.  A sign denotes each object in a set of objects.
+
# A sign denotes each object in a set of objects.
 +
# A sign denotes a set of objects.
   −
  2.  A sign denotes a set of objects.
+
The first option uses the notion of a set in a casual, informal, or metalinguistic way, and does not really commit us to the existence of sets in any formal way.  This is the more razoresque choice, much less risky, ontologically speaking, and so we may adopt it as our "nominal" starting position.
   −
The first option uses the notion of a set in a casual,
+
In this ''plural denotative'' component of the sign relation, we are looking at what may be seen as a functional relationship, in the sense that we have a piece of some function <math>f : O \to S,</math> such that <math>f(x_1) =\!</math> <math>f(x_2) =\!</math> <math>f(x_3) = y,\!</math> for example.  A function always admits of being factored into an "onto" (surjective) map followed by a "one-to-one" (injective) map, as discussed earlier.
−
informal, or metalinguistic way, and does not really
  −
commit us to the existence of sets in any formal way.
  −
This is the more razoresque choice, much less risky,
  −
ontologically speaking, and so we may adopt it as
  −
our "nominal" starting position.
     −
Now, in this "plural denotative" component of the sign relation,
+
But where do the intermediate entities go?  We could lodge them in a brand new space all their own, but Ockham the Innkeeper is right up there with Old Procrustes when it comes to the amenity of his accommodations, and so we feel compelled to at least try shoving them into one or another of the spaces already reserved.
−
we are looking at what may be seen as a functional relationship,
  −
in the sense that we have a piece of some function f : O -> S,
  −
such that f(x_1) = f(x_2) = f(x_3) = y, for example.  A function
  −
always admits of being factored into an "onto" (surjective) map
  −
followed by a "one-to-one" (injective) map, as discussed earlier.
     −
But where do the intermediate entities go?  We could lodge them
+
In the rest of this discussion, let us assign the label <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to the intermediate entity between the objects <math>x_j\!</math> and the sign <math>y.\!</math>
−
in a brand new space all their own, but Ockham the Innkeeper is
  −
right up there with Old Procrustes when it comes to the amenity
  −
of his accommodations, and so we feel compelled to at least try
  −
shoving them into one or another of the spaces already reserved.
     −
In the rest of this discussion, let us assign the label "i" to
+
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.
−
the intermediate entity between the objects x_j and the sign y.
  −
 
  −
Now, should you annex i to the object domain O 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%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L'  |
 
| Denotative Component of L'  |
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|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
   −
But if you assimilate i to the realm of signs S, you will
+
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.
−
be showing your inclination to remain within the straight
  −
and narrow of "Conceptualist" or even "Nominalist" dogmas,
  −
and you may read this "i" as standing for an intelligible
  −
concept, or an "idea" of the safely decapitalized, mental
  −
impression variety.
      +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
−
| Denotative Component of L"  |
+
| Denotative Component of L'' |
 
o--------------o--------------o
 
o--------------o--------------o
 
|  Objects    |    Signs    |
 
|  Objects    |    Signs    |
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|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
   −
But if you dare to be truly liberal, you might just find
+
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.
−
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%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
−
| Denotative Component of L'" |
+
| Denotative Component of L'''|
 
o--------------o--------------o
 
o--------------o--------------o
 
|  Objects    |    Signs    |
 
|  Objects    |    Signs    |
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|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
−
  −
To sum up, we have recognized the perfectly innocuous utility
  −
of admitting the abstract intermediate object i, that may be
  −
interpreted as an intension, a property, or a quality that
  −
is held in common by all of the initial objects x_j that
  −
are plurally denoted by the sign y.  Further, it appears
  −
to be equally unexceptionable to allow the use of the
  −
sign "i" to denote this shared intension i.  Finally,
  −
all of this flexibility arises from a universally
  −
available construction, a type of compositional
  −
factorization, common to the functional parts
  −
of the dyadic components of any relation.
   
</pre>
 
</pre>
 +
|}
   −
==Work Area==
+
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>x_j\!</math> that are plurally denoted by the sign <math>y.\!</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.
−
 
  −
<pre>
  −
The word "intension" has recently come to be stressed in our discussions.
  −
As I first learned this word from my reading of Leibniz, I shall take it
  −
to be nothing more than a synonym for "property" or "quality", and shall
  −
probably always associate it with the primes factorization of integers,
  −
the analogy between having a factor and having a property being one of
  −
the most striking, at least to my neo-pythagorean compleated mystical
  −
sensitivities, that Leibniz ever posed, and of which certain facets
  −
of Peirce's work can be taken as a further polishing up, if one is
  −
of a mind to do so.
  −
 
  −
As I dare not presume this to constitute the common acceptation
  −
of the term "intension", not without checking it out, at least,
  −
I will need to try and understand how others here understand
  −
the term and all of its various derivatives, thereby hoping
  −
to anticipate, that is to say, to evade or to intercept,
  −
a few of the brands of late-breaking misunderstandings
  −
that are so easy to find ourselves being surprised by,
  −
if one shies away from asking silly questions at the
  −
very first introduction of one of these parvenu words.
  −
I have been advised that it will probably be fruitless
  −
to ask direct questions of my informants in such a regard,
  −
but I do not see how else to catalyze the process of exposing
  −
the presumption that "it's just understood" when in fact it may
  −
be far from being so, and thus to clear the way for whatever real
  −
clarification might possibly be forthcoming, in the goodness of time.
  −
Just to be open, and patent, and completely above the metonymous board,
  −
I will lay out the paradigm that I myself bear in mind when I think about
  −
how I might place the locus and the sense of this term "intension", because
  −
I see the matter of where to lodge it in our logical logistic as being quite
  −
analogous to the issue of where to place those other i-words, namely, "idea",
  −
capitalized or not, "impresssion", "intelligible concept", and "interpretant".
  −
</pre>
      
==Document History==
 
==Document History==
    
===Nov 2000 &mdash; Factorization Issues===
 
===Nov 2000 &mdash; Factorization Issues===
−
  −
'''Ontology List'''
  −
  −
* http://suo.ieee.org/ontology/thrd111.html#00007
  −
# http://suo.ieee.org/ontology/msg00007.html
  −
# http://suo.ieee.org/ontology/msg00025.html
  −
# http://suo.ieee.org/ontology/msg00032.html
      
'''Standard Upper Ontology'''
 
'''Standard Upper Ontology'''
Line 494: Line 392:  
# http://suo.ieee.org/email/msg02430.html
 
# http://suo.ieee.org/email/msg02430.html
 
# http://suo.ieee.org/email/msg02448.html
 
# http://suo.ieee.org/email/msg02448.html
 +
 +
'''Ontology List'''
 +
 +
* http://suo.ieee.org/ontology/thrd111.html#00007
 +
# http://suo.ieee.org/ontology/msg00007.html
 +
# http://suo.ieee.org/ontology/msg00025.html
 +
# http://suo.ieee.org/ontology/msg00032.html
    
===Mar 2001 &mdash; Factorization Flip-Flop===
 
===Mar 2001 &mdash; Factorization Flip-Flop===
Jon Awbrey
12,089

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