SUO: Factorization Issues

[Jon Awbrey <jawbrey@oakland.edu>]

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CG & SUO Groups,

I would like to introduce a concept that I find to be of
use in discussing the problems of hypostatic abstraction,
reification, the reality of universals, and the questions
of choosing among nominalism, conceptualism, and realism,
generally.

I will first take this up in the simplest possible setting,
where it has to do with the special sorts of relations that
are called "functions", and after the basic idea is clear
in this easiest case I will deal with the more general
notion of "factorization" among, in, and of relations.

Picture an arbitrary function from a source (domain)
to a target (co-domain).  Here is one picture of an
f : X -> Y, just about as generic as it needs to be:

Source X = {1, 2, 3, 4,    5}
       |    .  .  .  .     .
   f   |     \ | /    \   /
       |      \|/      \ /
       v    .  .  .  .  .  .
Target Y = {A, B, C, D, E, F}

Now, it is a fact that any old function that you might
pick "factors" into a surjective ("onto") function and
an injective ("one-one") function, in my example, just
like this:

Source X = {1, 2, 3, 4,    5}
       |    .  .  .  .     .
   g   |     \ | /    \   /
       v      \|/      \ /
Middle M = {   b   ,    e   }
       |       |        |
   h   |       |        |
       v    .  .  .  .  .  .
Target Y = {A, B, C, D, E, F}

Writing functional compositions "on the right", as they say,
we have the following collection of data about the situation:

X = {1, 2, 3, 4, 5}
M = {b, e}
Y = {A, B, C, D, E, F}

f : X -> Y, arbitrary.
g : X -> M, surjective.
h : M -> Y, injective.

f = gh

What does this have to do with reification and such?
Well, suppose that the Source is a set of "objects"
and that the Target is a set of "signs".  I have to
break now, so I think that I will leave this as an
exercise for the reader until I can get back to it
later on tonight.

Cheers,

Jon Awbrey

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