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Directory:Jon Awbrey/Notes/Factorization Issues (view source)
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==Nominalism and Realism==
==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>
o---------o---------o---------o
o---------o---------o---------o
| Object | Sign | Interp |
| Object | Sign | Interp |
| ... | 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 |
| 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 |
| |
| |
o-----------------------------o
o-----------------------------o
</pre>
|}
I would like to -- but my personal limitations in the
I would like to — 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 |
| |
| |
o-----------------------------o
o-----------------------------o
</pre>
|}
Notice the subtle distinction between these two cases:
Notice the subtle distinction between these two cases:
# A sign denotes each object in a set of objects.
# 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.
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.
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.
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>
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.
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' |
| |
| |
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 |
| |
| |
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 |
| |
| |
o-----------------------------o
o-----------------------------o
</pre>
|}
To sum up, we have recognized the perfectly innocuous utility
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.
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.
==Work Area==
==Work Area==