Pre-mathematical Counting
Every concept articulated in language begins as a metaphor. Then the
poetic vitality associated with new metaphors wears away until the
metaphor dies and a taken-for-granted concept is petrified in its
place. The fact that even scientific concepts begin as metaphors has
come as a revelation to philosophers. It always seemed self-evident
that metaphor could have no place in science. How could any statement
that says one thing ("love is a rose") but means another ("love is
beautiful, many-layered, subtle, sometimes painful, and needs
nourishment") be scientific?
Ballard (1978, pp. 186-190) observes that "The need for enlarging language beyond the level of the literal invades even mathematics. This need is encountered by anyone who seeks the meaning of (say) the number two used to `count' two concrete individuals. Socrates was the first to note the oddity in the fact that, though he and Cebes are each one, yet if they are juxtaposed, then somehow together they become two (Phaedo 96d). In what sense are they two?"
"....when we speak of two concrete individuals, `two' is not given a literal but a figurative sense. In order to conclude that Socrates and Cebes together form a (quantitative) group of two, the measurer must ignore the Socratic character of Socrates and the Cebean nature of Cebes.... Thus, the concrete `two' refers us to unlike component unities. We may call this kind of unit pre-mathematical, for it cannot be used in counting objects but only for referring to objects before abstraction from their unique being has been made."
This pre-mathematical counting, that takes place before generalization and abstraction, is the counting of marks of correct and incorrect responses, or steps on a rating scale, which constitute the qualitative, ordinal observations. It is from these that quantitative measures derive. For the purposes of designing a measurement system, we "act as if", "entertain the possibility that", "suspend our disbelief in the fiction that" what we are counting is some kind of "one" thing.
Most Psychosocial Measurement is Pre-mathematical
The problem with virtually all educational, psychological, and social
measurement is that the metaphorical fiction entertained in the
counting is simply assumed true. To transform the pre-mathematical to
the mathematical, though, it is necessary to do more than supply
oneself with mindless entertainment; the kind of entertainment
provided by measurement must be the sort described by Plato as the
public display of every individual's private tragedy. The fiction
must be true even if it did not happen in every literal detail to
everyone involved; when the fiction is tested and found to hold up as
a story in which we can all find ourselves, then we have created a
model on which we can base our relations, our society, and in which we
can trust and have faith. Michell (1990) speaks of the transformation
from the pre-mathematical to the mathematical as testing the
quantitative hypothesis; to employ the language that Wright (Wright &
Masters, 1982) frequently uses, we must wonder if there is some common
story told by the data that is shared by all of the persons measured
and that is given a voice by the questions asked.
In most approaches to measurement the unique voices of the persons measured are drowned in the chorus bawling the story. The chorus is usually more cacophonous than harmonious because the most common approaches are based on treating raw score summaries as though they are already measures. In these kinds of pre-measurement, the counts employed remain pre-mathematical, in Ballard's sense, because nothing has been done to justify abstracting and generalizing from unique individuals to numbers that are always and everywhere a matter of adding one more of the same. Such pre-measurement approaches take no steps towards defining a valid construct along a measurement continuum in order to produce a reproducible quantitative solution.
Since the Renaissance, the natural sciences have become mathematically oriented. This has required explicit transformation of the pre- mathematical into the mathematical. Consequently, though claiming to reject metaphor, they have been unable to avoid the metaphoric process in practice, and this has been fundamental to their success. Success has been defined, though, in terms of the capacity to dominate, manipulate, and control nature (and the metaphoric process). But the freshly blooming realization of metaphor's creative role in science comes at the same time that we are coming to terms with ecological notions involving co-existence, stewardship, and co-evolution.
The jarring rhetoric of science's rejection of metaphor has so permeated the thinking of social scientists that they have succeeded, against their own best interests in keeping metaphor out of their measurement practice. There is a need for recognizing and accepting the fact that measurement is a matter of making metaphors and creating new meanings. This goes beyond mere concerns of methodological rigor to something more basic, more useful, more interesting.
Individuality
Ballard (ibid.) says that quantitative "measurement is justified and
sophisticated only when reparation is made by recalling the uniqueness
of the person measured. Otherwise, the abstraction does him
violence." It is of some importance, then, that Rasch (1966a, b) was
explicit about giving voice to individuals and harmonizing their
voices into a chorus that sings the same song. The effect of Rasch's
measurement requirements is to let the character of the questions and
answers show itself. Tests and surveys of unknown construct validity
blindly and unjustifiably treat all persons measured and items
measuring as the same, when, of course, they are not, providing only a
bland, homogenized kind of white noise that may do nothing more than
fail to disturb the unexamined assumptions and prejudices of the
researcher. By building construct analysis into instrument
calibration, Rasch provides an indication of each individual's
uniqueness, enabling us to recognize each person's and each item's
special strengths and weaknesses.
Our humanity depends upon allowing each individual's uniqueness to stand as a constant reminder of the creative act that successful measurement is. Ballard asks whether humanity, which has succeeded so well in using measurement to dominating nature, will be able to do anything through the application of social measurement technologies, except dominate and imprison itself.
Rasch's probabilistic conjoint measurement models demand that pre- mathematical, dialogical, counting numbers never be mistaken for measuring numbers. Rasch-based fit statistics assess the consistency of the data from each person and item with the model. They provide a constant reminder of the individuality of everyone and every question contributing to the measurement effort. These features open the door to an authenticity in social measurement that allows people's abilities and attitudes to be what they are, not just what can be manipulated and controlled. This is a crucial criterion that must be met if we are to understand ourselves.
Do we have a vision of what it will mean for humanity to succeed at maintaining what Ballard (ibid.) calls a "lively recollection of the [pre-mathematical, dialogical] sense of number in its reference to the beings which they measure"? It may be that those who employ Rasch's (1977) separability theorem as a criterion in the transformation of pre-mathematical counts to metaphorically and mathematically sound measures are those most involved in working toward this vision.
Ballard, Edward G. 1978. Man and Technology: Toward the Measurement of a Culture. Pittsburgh: Duquesne University Press.
Michell Joel. 1990. An introduction to the logic of psychological measurement. Hillsdale, NJ: Lawrence Erlbaum.
Rasch, Georg. 1966a. An individualistic approach to item analysis. In Readings in Mathematical Social Science. Ed. P. F. Lazarsfeld and N. W. Henry. Chicago: Science Research Associates.
Rasch, Georg. 1966b. An item analysis which takes individual differences into account. British Journal of Math and Stat Psychology, 19, 49-57.
Rasch, Georg. 1977. On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. Danish Yearbook of Philosophy, 14, 58-94.
Wright, Benjamin D. & Masters, Geofferey. 1982. Rating Scale Analysis. Chicago: MESA Press.
Counting as metaphor. Fisher WP Jr. … Rasch Measurement Transactions, 1994, 8:2 p.358
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