This is the main goal because measures of environmental variables affecting controlled perceptions are the fundamental data of PCT. [You said “all other behavioral research based on Control Theory”, but the distinction is not limited to CT research, correct?]
For reference, here is your Table 8.1 from that book:
For a perceptual variable to be described mathematically, must it first be quantified?
No. As an illustration of treating complex perceptual variables as mathematical objects, consider the book Mathematical structures of language. (At that page is a link to download a PDF image of the book.)
A more comprehensive treatment is in A theory of language and information. (Again, a PDF image of the book.)
The methodology to develop mathematical descriptions of complex controlled language variables employs the Test for Controlled Variables to identify perceptual variables as mathematical objects “in terms of the perceptual variables of which they are composed, variables that are themselves expressed mathematically in terms of the perceptual variables of which they are composed.” Obviously, there is a foundation of elementary data—there is no infinite regress into the imperceptible—but in the case of language these primitive data are themselves perceptual data.
The mathematics is from set theory and some linear algebra for mappings from subset to subset. Quantification can be brought in to the specification of data, e.g. the amount of difference in formant frequencies for segments of utterances which speakers perceive as contrasting (phonemically distinct), or some measure of the acceptability-rankings of homomorphous sentences, but such quantification is not essential; affirmations of perception suffice.
You identify the main goal (identifying what perceptions are controlled) and the main hypothesis (control cascades in a perceptual hierarchy). You do not mention the third principal distinction of PCT from other theories, which is a shame because it is one in which you have excelled: the construction of generative simulations which replicate observed behavior with fidelity that is unattainable by other theories of behavior.
It is for that purpose that quantified data have been seen as compulsory. Indeed, Bill, Dag, and others have emphasized that PCT is a ‘hard science’ like physics and chemistry because the PCT model is a quantitative model. But the higher we investigate in perceptual hierarchy, into complex perceptions which are of more general human interest and make for better PR and funding, quantification is more and more difficult.
What do we seek to quantify? PCT postulates neural signals, rates of firing in neurons which in principle can be measured (though they be averages across bundles of neurons); it postulates that these neural signals undergo amplification, damping, addition, subtraction, and other transformations between different parts of the nervous system; that they are transformed by ‘effectors’ into endocrine excretions, muscular efforts, and other physical effects in the body and in its environment; that sensory organs transform these effects in the internal and external environment back into neural signals entering the hierarchy at its lowest level, closing many concurrent control loops. In the theoretical model, all of these variables are quantities. In principal, all of these quantities can be measured.
So far, most PCT simulations (generative models of particular behavior) have been based on quantitative measurements of behavioral effects in the outer environment (efforts), environmental effects on sensors (stimuli) or as a surrogate measurements of environmental conditions detected by measuring instruments and determined by the Test to correspond to values of the controlled perceptual variable. Control loops that are closed through the interior corporeal environment are less accessible and have been much neglected in PCT
Precise quantification is not always necessary. In the CROWD demo, it does not matter precisely what the reference values for proximity to the ‘attractive’ agent and to fellow crowd members may be, only that these are controlled perceptions at some (moderate) reference level. The formation of rings and arcs is seen as a side effect of control, but in the wild it can also be a controlled perception as well (raconteur: “Gather round and hear my tale!”; teacher: “Circle time, children!”; policeman: “Disperse!”; participant: “Hm, people are gathered around her. Excuse me, can I squeeze in here?”).
The perceptual variables that constitute a language are not quantitative. The basic data, phonemic contrasts, may be thought of as categorial and the terms of their differentiation may be quantified along dimensions of their differentiation in several concurrent sensory modalities: for the articulation of speech, a combination of tactile and kinesthetic perceptions (from stretch and tension sensors), and auditory perceptions for the sounds of one’s own speech (always compared with that of others). There is a ‘quantal’ theory of speech (due to Ken Stevens, see here, here, and here) but the burden of it is not that speech sounds are quantities but rather that auditory perceptions are easier to control in certain parts of the acoustic space in which speech sounds are distinguished, that tactile and kinesthetic perceptions are easier to control in certain parts of the articulatory space in which speech organs differentiate them. This helps to explain that languages universally ‘prefer’ certain points of articulation and certain acoustic features (with a ranking of preference) despite that each of these is a location on indefinitely variable parameters.
People do not go around controlling individual phonemes. Even in the exceptional case where a syllable consists of a single phoneme (Mmmm!) it is an utterance that is controlled to be recognizably distinct from other possible utterances (Oooh! Ah! Eh!). The fundamental data of language are controlled perceptions of differences. Quantitatively, the degree of differentiation can drop even to nil if for other reasons no other word could occur in the linguistic and environmental context, and it is frequently the case that the given word is understood as present in the utterance even though not physically spoken (or written). An example is the noun which is the subject of the verb spoken and of the verb written in the preceding sentence. So this is another way in which the degree to which a given element of language contrasts to all other possible such elements in that position in the utterance is a controlled perception; as indeed is the perception of ‘such elements’ (elements of the same kind) and the perception of a position in an utterance with respect to its structure of kinds of elements in certain relationships. It comes down to relationships of dependency between words of different kinds (the kinds defined as requiring co-presence of a word or words of specified kinds) and the reductions due mostly to omitting overt expression of words whose presence can be reliably construed from context, but sometimes also socially institutionalized in arbitrary and historically contingent ways.
So the perceptions that constitute language are a quintessential example of collectively controlled perceptual variables; that is, people who use a given language control perceptions of correlation between the language perceptions that they control, and their reference values, with those which they perceive as being controlled by other users of that language.
A certain imprecision of control of the speech sounds one produces is a consequence of necessarily being unable to hear those sounds until it is too late to correct them, and the means of correction being the adjustment of reference values for tactile and kinesthetic perceptions which, at least consciously, feel rather vague and imprecise. This contributes to the great variability of observed and quantifiable acoustic outputs and articulatory targets and efforts toward them, variability which nonetheless is good enough for speaker and hearer to agree as to what words were spoken (sometimes with repetition).
Agreement as to the association of non-language perceptions with those utterances is another matter. Associative memory seems not to be a control process, though the establishment, strengthening or weakening, and above all the purposeful reconstruction of memory probably involves control processes of the midbrain. Bill’s concepts of a Category Level and his actual example in B:CP of what he considered Program Control (looking for his glasses in places that came to mind from memory as he moved around the room) involve this murky area that has been rather neglected in PCT.
In the above table and proposed database, perceptions at successively higher levels increasingly are controlled through environmental feedback paths which include features which are socially stabilized, or are themselves collectively controlled as in the case of language. This is only to acknowledge that we live in a built environment, that what is built must be maintained, and that humans rather than natural processes control the maintenance of human constructs. Collectively controlled perceptions are the invisible water in which we cosseted swim. The table and database must accommodate this.