Changes

==Note 1==

==Note 1==

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 one picture of an <math>f : X \to Y,</math> just about as generic as it needs to be:

to a Target (Co-domain).  Here is one picture of an

f : X -> Y, just about as generic as it needs to be:

o---------------------------------------o

o---------------------------------------o

|                                      |

|                                      |

|                                      |

|                                      |

o---------------------------------------o

o---------------------------------------o

Now, it is a fact that any old function that you might

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

pick "factors" into a surjective ("onto") function and

an injective ("one-to-one") function, in the present

example just like so:

o---------------------------------------o

o---------------------------------------o

|                                      |

|                                      |

|                                      |

|                                      |

o---------------------------------------o

o---------------------------------------o

Writing the functional compositions f = g o h "on the right",

Writing the functional compositions <math>f = g \circ h</math> "on the right", as they say, we have the following data about the situation:

as they say, we have the following data about the situation:

   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

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? Well, suppose that the source domain <math>X\!</math> is a set of ''objects'', that the target domain <math>Y\!</math> is a set of ''signs'', and suppose that the function <math>f : X \to Y</math> indicates the effect of a classification, conceptualization, discrimination, perception, or some other type of sorting operation, distributing the elements of the set <math>X\!</math> of

Well, suppose that the Source domain X is a set of "objects",

objects and into a set of sorting bins that are labeled with the elements of the set <math>Y,\!</math> regarded as a set of classifiers, concepts, descriptors, percepts, or just plain signs, whether these signs are regarded as being in the mind, as with concepts, or whether they happen to be inscribed more publicly in another medium.

that the Target domain Y is a set of "signs", and suppose that

the function f : X -> Y indicates the effect of a classification,

conceptualization, discrimination, perception, or some other type

of "sorting" operation, distributing the elements of the set X of

objects and into a set of "sorting bins" that are labeled with the

elements of the set Y, regarded as a set of classifiers, concepts,

descriptors, percepts, or just plain signs, whether these signs

are regarded as being in the mind, as with concepts, or whether

they happen to be inscribed more publicly in another medium.

In general, if we try to use the signs in the Target (Co-domain) Y

In general, if we try to use the signs in the Target (Co-domain) Y

to reference the objects in the Source (Domain) X, then we will be

to reference the objects in the Source (Domain) X, then we will be