12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Monday November 25, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Notes/Factorization And Reification (view source)
Revision as of 18:34, 24 June 2009
, 18:34, 24 June 2009→Factoring Functions: markup
==Factoring Functions==
==Factoring Functions==
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 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 take this up first in the simplest possible setting,
I will take this up first in the simplest possible setting, where it has to do with the special sorts of relations that are commonly called ''functions'', and after the basic idea is made as clear as possible in this easiest case I will deal with the notion of factorization as it affects more generic types of relations.
where it has to do with the special sorts of relations that
are commonly called "functions", and after the basic idea
is made as clear as possible in this easiest case I will
deal with the notion of "factorization" as it affects
more generic types of relations.
Picture an arbitrary function from a Source (Domain)
Picture an arbitrary function from a ''source'' or ''domain'' to a ''target'' or ''codomain''. Here is a picture of such function, <math>f : X \to Y,</math> as generic as it needs to be for our prsent purposes:
to a Target (Co-domain). Here is one picture of an
f : X -> Y, just about as generic as it needs to be:
<pre>
Source X = {1, 2, 3, 4, 5}
Source X = {1, 2, 3, 4, 5}
| o o o o o
| o o o o o
v o o o o o o
v o o o o o o
Target Y = {A, B, C, D, E, F}
Target Y = {A, B, C, D, E, F}
</pre>
It is a fact that any old function that you might pick "factors" into a functional composition of two other functions, a surjective ("onto") function and an injective ("one-to-one") function, in the present example pictured below:
pick "factors" into a surjective ("onto") function and
an injective ("one-to-one") function, in the present
example just like so:
<pre>
Source X = {1, 2, 3, 4, 5}
Source X = {1, 2, 3, 4, 5}
| o o o o o
| o o o o o
v o o o o o o
v o o o o o o
Target Y = {A, B, C, D, E, F}
Target Y = {A, B, C, D, E, F}
</pre>
Writing the functional compositions f = g o h "on the right",
Writing functional compositions <math>f = g \circ h</math> "on the right", we have the following data about the situation:
<pre>
X = {1, 2, 3, 4, 5}
X = {1, 2, 3, 4, 5}
M = {b, e}
M = {b, e}
f = g o h
f = g o h
</pre>
What does all of this have to do with reification and so on?
What does all of this have to do with reification and so on?
<pre>
Well, suppose that the Source domain X is a set of "objects",
Well, suppose that the Source domain X is a set of "objects",
that the Target domain Y is a set of "signs", and suppose that
that the Target domain Y is a set of "signs", and suppose that
