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Revision as of 01:46, 25 June 2009
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Now, at this stage of the game, if you ask: ''Is the object of the sign <math>y\!</math> one or many?'', the answer has to be: ''Not one, but many''. That is, there is not one <math>x\!</math> that <math>y\!</math> denotes, but only the three <math>x\!</math>'s in the object space. Nominal thinkers would ask: ''Granted this, what need do we have really of more excess?'' The maxim of the nominal thinker is ''never read a general name as a name of a general'', meaning that we should never jump from the accidental circumstance of a plural sign <math>y\!</math> to the abnominal fact that a unit <math>x\!</math> exists.
Now, at this stage of the game, if you ask: ''Is the object of the sign <math>y\!</math> one or many?'', the answer has to be: ''Not one, but many''. That is, there is not one <math>x\!</math> that <math>y\!</math> denotes, but only the three <math>x\!</math>'s in the object space. Nominal thinkers would ask: ''Granted this, what need do we have really of more excess?'' The maxim of the nominal thinker is ''never read a general name as a name of a general'', meaning that we should never jump from the accidental circumstance of a plural sign <math>y\!</math> to the abnominal fact that a unit <math>x\!</math> exists.
In actual practice this would be just one segment of a much larger sign relation, but let us continue to focus on just this one piece. The association of objects with signs is not in general a function, no matter which way, from <math>O\!</math> to <math>S\!</math> or from <math>S\!</math> to <math>O,\!</math> that we might try to read it, but very often one will choose to focus on a selection of links that do make up a function in one direction or the other.
In actual practice this would be just one segment of a much larger
sign relation, but let us continue to focus on just this one piece.
The association of objects with signs is not in general a function,
no matter which way, from O to S or from S to O, that we might try
to read it, but very often one will choose to focus on a selection
of links that do make up a function in one direction or the other.
In general, but in this context especially, it is convenient
In general, but in this context especially, it is convenient to have a name for the converse of the denotation relation, or for any selection from it. I have been toying with the idea of calling this ''annotation'', or maybe ''ennotation''.
to have a name for the converse of the denotation relation,
or for any selection from it. I have been toying with the
idea of calling this "annotation", or maybe "ennotation".
For a not too impertinent instance, the assignment of the
For example, the assignment of the general term <math>y</math> to each of the objects <math>x_1, x_2, x_3\!</math> is one such functional patch, piece, segment, or selection. So this patch can be pictured according to the pattern that was previously observed, and thus transformed by means of a canonical factorization.
general term y to each of the objects x_1, x_2, x_3 is
one such functional patch, piece, segment, or selection.
So this patch can be pictured according to the pattern
that was previously observed, and thus transformed by
means of a canonical factorization.
In this case, we factor the function f : O -> S
In our example of a sign relation, we had a functional subset of the following shape:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
o---------------------------------------o
| |
| Source O :> x_1 x_2 x_3 |
| | o o o |
| | \ | / |
| f | \ | / |
| | \|/ |
| v ... o ... |
| Target S :> y |
| |
o---------------------------------------o
</pre>
|}
into the composition g o h, where g : O -> M, and h : M -> S
The function <math>f : O \to S</math> factors into a composition <math>g \circ h,\!</math> where <math>g : O \to M,</math> and <math>h : M \to S,</math> as shown here:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
o---------------------------------------o
| |
| Source O :> x_1 x_2 x_3 |
| |
| | o o o |
| g | \ | / |
| |
| | \ | / |
| v \|/ |
| Medium M :> ... x ... |
| | | |
| h | | |
| | | |
| v ... o ... |
| Target S :> y |
| |
o---------------------------------------o
</pre>
|}
The factorization of an arbitrary function
The factorization of an arbitrary function into a surjective ("onto") function followed by an injective ("one-one") function is such a deceptively trivial observation that I had guessed that you would all wonder what in the heck, if anything, could possibly come of it.
into a surjective ("onto") function followed
by an injective ("one-one") function is such
a deceptively trivial observation that I had
guessed that you would all wonder what in the
heck, if anything, could possibly come of it.
<pre>
What it means is that, "without loss or gain of generality" (WOLOGOG),
What it means is that, "without loss or gain of generality" (WOLOGOG),
we might as well assume that there is a domain of intermediate entities
we might as well assume that there is a domain of intermediate entities
