Directory talk:Jon Awbrey/Notes/Factorization And Reification

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Let me 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 's'
is quite literally taken to denote each object o<j> in a whole
collection of objects {o<1>, ..., o<k>, ...}, a situation that
I would normally represent in a sign-relational table like so:

      Object     Sign     Interp
    o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
        o1        's'       ...
        o2        's'       ...
        o3        's'       ...
    o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

For the moment, it does not matter what the interpretants are.

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 the Sign columns of the table.

    o1 ------>
              \
               \
    o2 ------>--@ 's'
               /
              /
    o3 ------>

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:

    o   o   o >>>>>>>>>>>>> 's'

Notice the subtle distinction between these two cases:

1.  A sign denotes each object in 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 starting position.

Now, 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 ahve a piece of some function f : O -> S,
such that f(o1) = f(o2) = f(o3) = 's', for example.  Functions
always admit of factoring into an "onto" (no relation) map and
then a one-one map, as we discussed what seems like an age ago.

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 give the name "i" to the
intermediate entity between the objects o<j> and the sign 's'.

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
may as well have the comforts of your Ideals in your exhile.

        i
       /|\   *
      / | \       *
     /  |  \           *
    o   o   o >>>>>>>>>>>> "s"

But if you assimilate i to the realm of signs S, you will
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.

    o   o   o >>>>>>>>>>>> "s"
        .  .  .             |
             . . .          |
                  ...       |
                       .    |
                           "i"

Note on Intension

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.

Previous Discussion

JW = Jim Willgoose

Re: FAR 2.  http://stderr.org/pipermail/inquiry/2005-May/thread.html#2747
In: FAR.    http://stderr.org/pipermail/inquiry/2005-May/002748.html

JW: What does "middle m" do? It appears to simplify the object domain.
    But to what end?  You could be a hardcore reductionist and allow full
    reality to source and middle (providing middle m has a role).  Gaining
    clarity doesn't compromise realism, even realism about sets (so long as
    they can be identified).  One can even be a Platonic nominalist (demanding
    that abstract objects be identified and individuated) and preserve a sense
    of realism by arguing that the identity and individuating character of an
    abstract object is independant of you, me etc.  This could be extended to
    possible uses for middle m that have not been discovered.  There are a lot
    of ways to hang on to ONE THING, champion reduction, and preserve realism.

I am here making use of a simple theorem from mathematical category theory,
the fact that any function can be factored into a surjective part followed
by an injective part, to provide the grounds for a compromise between some
ancient philosophical combatants.  Of course, the more reductionist among
those parties would no more warm up to the reality of functions than they
take kindly to the existence of those abstract objects we call "sets",
but that is neither here nor there.

Another point of the exercise was to examine the waffle room that we often
find in regard to the "connotations" of signs, whether they are more like
interpretant signs or more like intensions considered as abstract objects.
I think that I have at least outlined a way that we can have our cake and
eat it too.

The rest of your statements are very puzzling to me.
I can only guess that you've never met any hardcore
reductionists, as they certainly don't tolerate the
existence of sets, considered as something over and
above their individual elements.  And I have no way
of conjuring up what a platonic nominalist might be.