Thomas Johnson wrote, "The parabola represents what we know at the scientific level of abstraction ...". This is simply "wrong" or naive general semantics. Anything we say (about what the parabola represents) is a projection of our high-level abstract models onto what is going on. The parabola represents that from which our lowest levels of abstraction are abstracted from - what Bois and others have called "what is going on" (WIGO). The "zero" level of abstraction is not itself an abstraction, but ALL our "experiences" are abstraction responses at levels higher than "zero". Anything we experience at the silent levels; anything we experience perceptually, anything we verbalize, etc., including all our statements about "mad dance of electrons" as well as using the phrase WIGO itself are NOT what the phrase WIGO or the parabola represent - the level BEFORE any abstraction.

We must distinguish very carefully through the mechanism of consciousness of abstracting that anything we experience or say, including theories as to what the WIGO "is like", are maps and NOT the "base" or "bottom-level" territory.

The parabola represents what is going on that we cannot directly know or experience, because every experience, theory, or model, is an abstraction and a semantic reaction that is NOT "IT" (as Pula and others used to say at Seminars).

The "circularity of human knowledge", represented in structural differential by the dashed arrow leading back up and pointing at the parabola, indicates how the models we construct in both science as well as even at non-verbal levels in our nervous system, influences our abstracting. It shows that we project our models, beliefs, etc., onto WIGO. Our best current explanation accounts for our experiences, but it is only the map, as is every scientific theory, every perception, every individual brain experience, and as such is a map, it is NOT the territory (WIGO). What science "knows" is its own map, and this map is an interpretation of our collective verbal level experiences, but it is NOT WIGO. What science "knows" is a nice packaged children's story to account to our simple minds for our experiences.

What science "knows" has changed many times over the millennia, and each time we "think" we have attained a "true" picture of WIGO, it turns out, sooner or later, that it was wrong. Each major scientific paradigm has been overthrown. If Kuhn and general semantics teaches us anything, it teaches us that our so-called "knowledge" is only tentative and conditional until it is revised, and revised, and revised, from paradigm to paradigm. So the parabola cannot represent any scientific "knowledge"; an open ended parabola indicating an infinity of possibilities from which to abstract is exactly and only what Korzybski intended.

Science does not "know" WIGO; Science builds a model - actually many models - to account for our abstractions. Fortunately, however, many small details of our model have not had to be revised in a while, so we have a sense of dependable constancy that can give us a sense of security and a false sence that we "know" something about WIGO.

The parabola represents only WIGO. Our so-called "knowledge" is represented in the object level (non-verbal) and the label tags (verbal levels). The dashed arrow indicates that we abstract into those pre-conceieved ideas, or occasionaly, alter them.

Circularity of human knowledge is established in the dashed arrow pointing back to the parabola. This is NOT to be interpreted that we can know what is going on; we cannot; we can only "know" our abstractions. This places all and any scientific "knowledge" squarely in the upper levels of abstracting, not in the parabola.

"Predication"? Interesting slip.

The volume of generated formulations bears an inverse relation to the clarity of a stimulus.

Note above how much and how varied the stimulus "Structure is the sole content of knowledge." has generated.

"Structure", "Order", and "relation" have been characterized as "undefined" terms analogous to "point", "line", and "plane" in geometry. Each can be defined in terms of the others.

"Order" is a relation between structures.
A "relation" is an ordering of structures.
A "structure" is an ordered relation.

a "relation" on the Cartesian product of X and Y is defined as a set of ordered pairs (x,y) where x is in or from X and y is in or from Y. This is an example of defining "relation" in terms of order and structure.

Note that order, when related to time becomes process - a sequence of events.

The difference between a relation and a function, is that a function can only have one outcome for a given set of conditions.

Consider the function "twice". Given a particular input, you always get the same output.
"Twice" 6 is always 12.

Consider the relation "bigger". Given a particular input, say 6, you can get many different outputs. 7, 8, 9, etc. are all "bigger" than 6.

Twice is a function, bigger is a relation.

Many (most?) lay-persons fail to make this precise distinction.

As I've noted elsewhere, now-a-days I prefer to differentiate between "information" and "knowledge" in the following manner.

1. "Information" is encoded in symbols and transmitted from generation to generation in what is called (in general semantics circles) "time-binding".

2. "Knowledge", on the other hand, is the ability to use information, and that must be learned anew by each individual in the time-binding population.

This defining difference can be found as becoming fairly standard in industry in the area known as "knowledge management", a field I worked in, while at Keane, Inc., as an Information Technology specialist.

As such, "knowledge" may be verbal or non-verbal; it can also be conscious or unconscious.

"Structure is the sole content of knowledge" needs to be expanded. When we think we "know" something, we have a cognitive model that we use to navigate with - information and the ability to use that information. When information is used, it guides actions, and actions have effects that result in new sensory experiences for the actor.

The typical sense of "struture" is a recursive notion that involves "parts" that are assembled into a whole involving relations among the parts. This can be the components of an object or it can be the sequence of event-actions in a plan or record of actions.

The realist perspective assumes a correspondence between the "structure" of the model and "strucure" of that which is modeled, hence realists talk about "similarity of structure".

A very charitable interpretation of "Structure is the sole content of knowledge" would attend to the model without assuming that the model "corresponded with similarity" to something other than the model.

I have my cognitive (conscious and subconscious) maps, and I use them to determine what my next action will be. I respond to the abstractions and semantic reactions I experience as a result of my prior actions. My maps have "structure". My criteria for effectiveness of my maps is that the less often I need to revise my maps, the better I evaluate them to be. When I have maps or models that have not been revised over many uses, I label those maps "knowledge" and say I know how to use the information in question.

Another volume of formulations.
Perhaps some useful to some.
Nuff said.

In order...
Question 1.
We understand order metaphorically in terms of our ability to move from one thing to another. All three of "structure", "order", and "relation" are abstract responses to a lower level of abstraction. "Divisibility" itself respresents a capability of our abstracting process. So that which is abstracted as divided was undivided prior to abstraction. So structure and process (order) are divided or not divided according to the level of abstraction of the observer.

Question 2.
No. A function has a domain - the independent variable(s) - and a range - the dependent variable. "Unique" contexts represent different points in the domain, and a function can have different values at different points. It cannot have different values at the same point. Only a constant function would have the same value at different points.

Question 3.
Why should I choose?, said the philosopher.

If we take the general semantics principle of abstracting seriously, we only know our models, because there is no independent way of validating them. No matter how many times we test our models, there is always a next time, and then we might find an error. This perspective has been around for a lot longer than general semantics; it has permeated philosophy for over two and a half millennia, although it was lost during the dark ages and the cultures dominated by religion.

See Heraclitus? or Xenophanes?

For everyday activites the perspective of realism seems to work reasonably well. We have lots of models that have not failed us in a long time. But for the really big questions, we don't know.

I am using my not-yet-disconfirmed model to navigate and predict my next experiences. My model has a structure. If I assume that the structure of my model is like the impression in the wax from a signet ring - carries away its form - then I'm stepping outside what can be proven - and into the domain of all the other religions - assuming what cannot be proven.

There is metaphysics - theories of being or what can exist - and epistemology - theories of how we can know. General semantics is "modern, open, applied, epistemology", and as such cannot delve into metaphysics. To be a "realist" one must assume that "stuff" exists and that it has structure independently of how we can know about it. According to general semantics we can only know our models; we cannot know about existing "stuff" any more than that we abstract from it. This does NOT mean that our abstractions "are" (being - metaphysics) characteristics of that "stuff" (WIGO). Abstractions are responses - effects - not the causes. We already "know" that our perception of color does not correspond to properties inherent in the physical world (as our model describes it).

Our visual system abstracts moving trapezoidal and curved images, but our brains turn those into fixed rectangular and straight objects of perception. What can we "know" about the cause of the object level visual system abstractions? We model them. We don't "know" them.

Do I support or reject the realist perspective? "yes."
Do I support and reject the realist perspective? "yes."

Thomas wrote, I think it would be more accurate to say that our models are never complete because the nature of abstraction precludes the possibility of considering all factors. So there will always be some discrepancy between our models and our measurements and it really boils down to reducing this as time goes on. Many are still in use because the added accuracy of newer models is not required, ie. classic vs relativistic physics.

1. "Accuracy" as you are using it presumes realism and that there is some way other than abstraction to determine the degree to which an abstraction "matches" that which it is abstracted from; it consfuses metaphysics and epistemology. It fails to recognize that we cannot get to the "zero" level of abstraction. You also held up the notion of a "complete model" as somehow having no descrepancies between the model and that which it models. This is contrary to general semantics, because the model (map) is not the territory, covers not all the territory, and has errors. The ONLY way a "model" can have zero differences is for it to be identical to the territory; that is to say, not a model (map), but the thing itself. If you really want to get "inside" general semantics you must let go of the realist perspective, drop the "god's eye view" that presumes it's possible to evaluate, independently of abstracting, how well a map or model "reflects" the territory. "Accuracy" is a realism based notion that presumes to compare a measurement deviation to the actual thing itself.

"Precision" is a notion that measures the repeatability of a measure with small variation.

A measure can be both precise and "inaccurate".
If we have a "standard" measure from which others are compared, then we would call a precise measurement that agrees with the standard measurement "accurate". In the case of abstraction we do not have a "standard" abstraction to provide the reference point.

WE CANNOT GET OUTSIDE OUR ABSTRACTIONS, PERIOD!
I have my model, which happens to be the product of my abstracting process. I navigate with that model, making predictions as to what I will subsequently abstract. ONLY when my prediction does not come to pass, do I discover that something is wrong with my model. I then change my model, and then continue to use it until another prediction fails. This way of describing the events does not assume realism. It does not take a metaphysical perspective. It is strictly an "inside" perspective. I only know my model, and I evaluate my model (and submodels) as "good" in proportion to how seldom I need to modify them.

Realists project models onto WIGO and assume that the structure of our models "exists" "out there" in the WIGO. They identify their models as being structures "in the world". But every structure we think we know is our mental construct projected - the brain locating its experiences elsewhere.

David,
Functions are mathematical. Y=2 is an example of a constant function. What do you mean by "real life"? If you wish to refer to something that we might use a constant function to represent, you are talking not about a function, but about that from which a function is an abstraction performed by a nervous system.

Suppose you call a company with many phone numbers (the domain) in the evening (after hours), and, no matter what number you dial, you always get the same recording. "The office is closed, call during business hours." This is an example that maps to a constant function in which all phone calls map to a single value. In the day time, the operator flips a switch on the pbx, and incoming calls are directed to different offices (numbers) based on the number dialed. Each of several numbers goes to its own office, and every time you dial a particular number, you get the same office. Different numbers dialed go to different offices. Sometimes a couple of numbers go to the same office, but each time you dial one, you always get the same result. This instantiates a function. For any particular number in the range of dialed numbers, you always get the same office.

Compare that with the big companies with many customer service representatives. Each time you call in, you might get a different customer service representative. One dialed number can map to several different desks. This illustrates a relation. It is not a function in the strict sense. Several different numbers in the company each go to several customer service representative desks. It's a many to many relation, not a function. Function always get the same result for the same input. Both functions and relations can get different results for different inputs.