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| | 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>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. |
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| − | ==Work Area==
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| − | ===Work Note 1===
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| − |
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| − | <pre>
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| − | Let me illustrate what I think that a lot of our controversies
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| − | about nominalism versus realism actually boil down to in practice.
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| − | From a semiotic or a sign-theoretic point of view, it all begins
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| − | with a case of "plural reference", which happens when a sign 's'
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| − | is quite literally taken to denote each object o<j> in a whole
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| − | collection of objects {o<1>, ..., o<k>, ...}, a situation that
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| − | I would normally represent in a sign-relational table like so:
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| − |
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| − | Object Sign Interp
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| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | o1 's' ...
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| − | o2 's' ...
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| − | o3 's' ...
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| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − |
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| − | For the moment, it does not matter what the interpretants are.
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| − |
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| − | I would like to diagram this somewhat after the following fashion,
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| − | here detailing just the denotative component of the sign relation,
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| − | that is, the 2-adic relation that is obtained by "projecting out"
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| − | the Object and the Sign columns of the table.
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| − |
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| − | o1 ------>
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| − | \
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| − | \
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| − | o2 ------>--@ 's'
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| − | /
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| − | /
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| − | o3 ------>
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| − |
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| − | I would like to -- but my personal limitations in the
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| − | Art of ASCII Hieroglyphics do not permit me to maintain
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| − | this level of detail as the figures begin to ramify much
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| − | beyond this level of complexity. Therefore, let me use
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| − | the following device to symbolize the same configuration:
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| − |
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| − | o o o >>>>>>>>>>>>> 's'
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| − |
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| − | Notice the subtle distinction between these two cases:
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| − |
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| − | 1. A sign denotes each object in a set of objects.
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| − |
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| − | 2. A sign denotes a set of objects.
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| − |
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| − | The first option uses the notion of a set in a casual,
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| − | informal, or metalinguistic way, and does not really
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| − | commit us to the existence of sets in any formal way.
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| − | This is the more razoresque choice, much less risky,
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| − | ontologically speaking, and so we may adopt it as
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| − | our starting position.
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| − |
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| − | Now, in this "plural denotative" component of the sign relation,
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| − | we are looking at what may be seen as a functional relationship,
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| − | in the sense that we ahve a piece of some function f : O -> S,
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| − | such that f(o1) = f(o2) = f(o3) = 's', for example. Functions
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| − | always admit of factoring into an "onto" (no relation) map and
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| − | then a one-one map, as we discussed what seems like an age ago.
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| − |
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| − | But where do the intermediate entities go? We could lodge them
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| − | in a brand new space all their own, but Ockham the Innkeeper is
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| − | right up there with Old Procrustes when it comes to the amenity
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| − | of his accommodations, and so we feel compelled to at least try
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| − | shoving them into one or another of the spaces already reserved.
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| − |
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| − | In the rest of this discussion, let us give the name "i" to the
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| − | intermediate entity between the objects o<j> and the sign 's'.
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| − |
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| − | Now should you annex i to the object domain O, you will have
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| − | instantly given yourself away as having "Realist" tendencies,
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| − | and you might as well go ahead and call it an "intension" or
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| − | even an "Idea" of the grossly subtlest Platonic brand, since
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| − | you are about to booted from Ockham's Establishment, and you
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| − | may as well have the comforts of your Ideals in your exhile.
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| − |
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| − | i
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| − | /|\ *
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| − | / | \ *
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| − | / | \ *
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| − | o o o >>>>>>>>>>>> "s"
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| − |
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| − | But if you assimilate i to the realm of signs S, you will
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| − | be showing your inclination to remain within the straight
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| − | and narrow of "Conceptualist" or even "Nominalist" dogmas,
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| − | and you may read this "i" as standing for an intelligible
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| − | concept, or an "idea" of the safely decapitalized, mental
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| − | impression variety.
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| − |
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| − | o o o >>>>>>>>>>>> "s"
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| − | . . . |
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| − | . . . |
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| − | ... |
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| − | . |
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| − | "i"
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| − | </pre>
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| − |
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| − | ===Work Note 2===
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| − |
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| − | The word ''intension'' has recently come to be stressed in our discussions. As I first learned this word from my reading of Leibniz, I shall take it to be nothing more than a synonym for ''property'' or ''quality'', and shall probably always associate it with the primes factorization of integers, the analogy between having a factor and having a property being one of the most striking, at least to my neo-pythagorean compleated mystical sensitivities, that Leibniz ever posed, and of which certain facets of Peirce's work can be taken as a further polishing up, if one is of a mind to do so.
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| − |
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| − | As I dare not presume this to constitute the common acceptation of the term ''intension'', not without checking it out, at least, I will need to try and understand how others here understand the term and all of its various derivatives, thereby hoping to anticipate, that is to say, to evade or to intercept, a few of the brands of late-breaking misunderstandings that are so easy to find ourselves being surprised by, if one shies away from asking silly questions at the very first introduction of one of these parvenu words. I have been advised that it will probably be fruitless to ask direct questions of my informants in such a regard, but I do not see how else to catalyze the process of exposing the presumption that "it's just understood" when in fact it may be far from being so, and thus to clear the way for whatever real clarification might possibly be forthcoming, in the goodness of time. Just to be open, and patent, and completely above the metonymous board, I will lay out the paradigm that I myself bear in mind when I think about how I might place the locus and the sense of this term ''intension'', because I see the matter of where to lodge it in our logical logistic as being quite analogous to the issue of where to place those other i-words, namely, ''idea'', capitalized or not, ''impresssion'', ''intelligible concept'', and ''interpretant''.
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| − |
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| − | ==Discussion==
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| − |
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| − | <pre>
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| − | JW = Jim Willgoose
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| − |
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| − | Re: FAR 2. http://stderr.org/pipermail/inquiry/2005-May/thread.html#2747
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| − | In: FAR. http://stderr.org/pipermail/inquiry/2005-May/002748.html
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| − |
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| − | JW: What does "middle m" do? It appears to simplify the object domain.
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| − | But to what end? You could be a hardcore reductionist and allow full
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| − | reality to source and middle (providing middle m has a role). Gaining
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| − | clarity doesn't compromise realism, even realism about sets (so long as
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| − | they can be identified). One can even be a Platonic nominalist (demanding
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| − | that abstract objects be identified and individuated) and preserve a sense
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| − | of realism by arguing that the identity and individuating character of an
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| − | abstract object is independant of you, me etc. This could be extended to
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| − | possible uses for middle m that have not been discovered. There are a lot
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| − | of ways to hang on to ONE THING, champion reduction, and preserve realism.
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| − |
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| − | I am here making use of a simple theorem from mathematical category theory,
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| − | the fact that any function can be factored into a surjective part followed
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| − | by an injective part, to provide the grounds for a compromise between some
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| − | ancient philosophical combatants. Of course, the more reductionist among
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| − | those parties would no more warm up to the reality of functions than they
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| − | take kindly to the existence of those abstract objects we call "sets",
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| − | but that is neither here nor there.
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| − |
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| − | Another point of the exercise was to examine the waffle room that we often
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| − | find in regard to the "connotations" of signs, whether they are more like
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| − | interpretant signs or more like intensions considered as abstract objects.
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| − | I think that I have at least outlined a way that we can have our cake and
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| − | eat it too.
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| − |
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| − | The rest of your statements are very puzzling to me.
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| − | I can only guess that you've never met any hardcore
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| − | reductionists, as they certainly don't tolerate the
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| − | existence of sets, considered as something over and
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| − | above their individual elements. And I have no way
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| − | of conjuring up what a platonic nominalist might be.
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| − | </pre>
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| | ==Document History== | | ==Document History== |