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==Nominalism and Realism==

==Nominalism and Realism==

Let me now 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 <math>y\!</math> is quite literally taken to denote each object <math>x_j\!</math> in a whole collection of objects <math>\{ x_1, \ldots, x_k, \ldots \},</math> a situation that can be represented in a sign-relational table like this one:

Let me now 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 <math>s\!</math> is quite literally taken to denote each object <math>o_j\!</math> in a whole collection of objects <math>\{ o_1, \ldots, o_k, \ldots \},</math> a situation that can be represented in a sign-relational table like this one:

{| align="center" cellspacing="10" style="text-align:center; width:90%"

{| align="center" cellspacing="10" style="text-align:center; width:90%"

| Object  |  Sign  | Interp  |

| Object  |  Sign  | Interp  |

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

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

|  x_1   |    y   |  ...  |

|  o_1   |    s   |  ...  |

|  x_2   |    y   |  ...  |

|  o_2   |    s   |  ...  |

|  x_3   |    y   |  ...  |

|  o_3   |    s   |  ...  |

|  ...  |    y   |  ...  |

|  ...  |    s   |  ...  |

|  x_k   |    y   |  ...  |

|  o_k   |    s   |  ...  |

|  ...  |    y   |  ...  |

|  ...  |    s   |  ...  |

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

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

</pre>

</pre>

| Object  |  Sign  | Interp  |

| Object  |  Sign  | Interp  |

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

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

|  x_1   |    y   |  ...  |

|  o_1   |    s   |  ...  |

|  x_2   |    y   |  ...  |

|  o_2   |    s   |  ...  |

|  x_3   |    y   |  ...  |

|  o_3   |    s   |  ...  |

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

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

</pre>

</pre>

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

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

|                            |

|                            |

|  x_1 o------>              |

|  o_1 o------>              |

|              \            |

|              \            |

|                \            |

|                \            |

|  x_2 o------>--o  y       |

|  o_2 o------>--o  s       |

|                /            |

|                /            |

|              /            |

|              /            |

|  x_3 o------>              |

|  o_3 o------>              |

|                            |

|                            |

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

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

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

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

|                            |

|                            |

| o  o  o >>>>>>>>>>>> y   |

| o  o  o >>>>>>>>>>>> s   |

|                            |

|                            |

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

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

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 "nominal" starting position.

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 "nominal" starting position.

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 have a piece of some function <math>f : O \to S,</math> such that <math>f(x_1) =\!</math> <math>f(x_2) =\!</math> <math>f(x_3) = y,\!</math> for example.  A function always admits of being factored into an "onto" (surjective) map followed by a "one-to-one" (injective) map, as discussed earlier.

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 have a piece of some function <math>f : O \to S,</math> such that <math>f(o_1) =\!</math> <math>f(o_2) =\!</math> <math>f(o_3) = s,\!</math> for example.  A function always admits of being factored into an "onto" (surjective) map followed by a "one-to-one" (injective) map, as discussed earlier.

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.

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 assign the label <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to the intermediate entity between the objects <math>x_j\!</math> and the sign <math>y.\!</math>

In the rest of this discussion, let us assign the label <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to the intermediate entity between the objects <math>o_j\!</math> and the sign <math>s.\!</math>

Now, should you annex <math>i\!</math> to the object domain <math>O\!</math> 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 might as well have the comforts of your ideals in your exile.

Now, should you annex <math>i\!</math> to the object domain <math>O\!</math> 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 might as well have the comforts of your ideals in your exile.

|  / | \      *            |

|  / | \      *            |

|  /  |  \          *        |

|  /  |  \          *        |

| o  o  o >>>>>>>>>>>> y   |

| o  o  o >>>>>>>>>>>> s   |

|                            |

|                            |

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

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

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

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

|                            |

|                            |

| o  o  o >>>>>>>>>>>> y   |

| o  o  o >>>>>>>>>>>> s   |

|    .  .  .            '    |

|    .  .  .            '    |

|        . . .          '    |

|        . . .          '    |

|  / | \      *            |

|  / | \      *            |

|  /  |  \          *        |

|  /  |  \          *        |

| o  o  o >>>>>>>>>>>> y   |

| o  o  o >>>>>>>>>>>> s   |

|    .  .  .            '    |

|    .  .  .            '    |

|        . . .          '    |

|        . . .          '    |

|}

|}

To sum up, we have recognized the perfectly innocuous utility of admitting the abstract intermediate object <math>i,\!</math> that may be interpreted as an intension, a property, or a quality that is held in common by all of the initial objects <math>x_j\!</math> that are plurally denoted by the sign <math>y.\!</math>  Further, it appears to be equally unexceptionable to allow the use of the sign <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to denote this shared intension <math>i.\!</math>  Finally, all of this flexibility arises from a universally available construction, a type of compositional factorization, common to the functional parts of the 2-adic components of any relation.

To sum up, we have recognized the perfectly innocuous utility of admitting the abstract intermediate object <math>i,\!</math> that may be interpreted as an intension, a property, or a quality that is held in common by all of the initial objects <math>o_j\!</math> that are plurally denoted by the sign <math>s.\!</math>  Further, it appears to be equally unexceptionable to allow the use of the sign <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to denote this shared intension <math>i.\!</math>  Finally, all of this flexibility arises from a universally available construction, a type of compositional factorization, common to the functional parts of the 2-adic components of any relation.

==Document History==

==Document History==