Plato's Separability Theorem

What does Gadamer (1989, p. 412) mean when he says that "We see that it is not word but number that is the real paradigm of the noetic"? The noetic, from the ancient Greek noesis, is the entire abstract population of all things that can be understood. Gadamer is addressing the same issue taken up by Descartes (1961, p. 8) when he says that "...in seeking the correct path to truth we should be concerned with nothing about which we cannot have a certainty equal to that of the demonstrations of arithmetic and geometry."

Mathematical thinking is too often assumed to be inherently numerical and quantitative (Michell 1990, 1999). The mere use of numbers in many fields is deemed sufficient indication of mathematical thinking, though the numbers may only rarely express anything substantively meaningful.

So how could Gadamer, the quintessential hermeneutic philosopher and anti-methodologist, seriously consider number to be the paradigmatic model of understandable meaning? He gives a vital clue when he acknowledges that "numerical signs [are coordinated] with particular numbers, and they are the most ideal signs because their position in the order completely exhausts them" (Gadamer 1989, p. 413), and he elsewhere gives extensive consideration to the "overall structural parallel between number and logos" (Gadamer 1980, p. 149; also see Gadamer 1979), but for those familiar with Rasch's separability theorem, perhaps no one illustrates the crux of the situation better than Ballard (1983).

To see the value in Ballard's treatment, first recall Rasch's (1961, p. 325) statement of the separability theorem: "On the basis of [one of the equations in the model] we may estimate the item parameters independently of the personal parameters, the latter having been replaced by something observable, namely, by the individual total number of correct answers. Furthermore, on the basis of [the next equation] we may estimate the personal parameters without knowing the item parameters which have been replaced by the total number of correct answers per item. Finally, [the third equation] allows for checks on the model [another equation] which are independent of all the parameters, relying only on the observations."

Ballard is concerned with Plato's examination of the concepts of small and large in the Republic (523D-525D). Plato shows that these concepts are insufficient for rigorous comparison due to the ambiguity of having things that can simultaneously be smaller than some things and larger than others. Ballard (1983, pp. 135-6) points out that Plato has Socrates "show that the confusion engendered by a finger being both large and small may be cleared up by the art of quantitative measurement. In order to execute this measurement, we first separate each finger from one of its properties, in this instance a quantitative property, its length, and replace each finger by its (abstract) length. So now the length of each finger can be measured by some equal and common unit of length; then one of these fingers, the middle, will appear to be larger by so much (i.e., by so many units of length) than another, and smaller by so much ...."

Later, Ballard (1983, pp. 136) points out that Plato also sees to it that "an analogous technique is brought to bear upon the puzzling aspect of the unit and the techniques of measure," so that "a still greater clarification can be achieved." So in the same way that Socrates separates the concrete instance of the thing to be measured from its quantitative property, and also separates the unit and the techniques of measurement from their concrete expressions, Rasch separates observations from parameters for both person measures and item calibrations.

Gadamer (1980, p. 150) takes the matter still further, pointing out that a "characteristic of a proportion is that its mathematical value is independent of the given factors in it, provided that they keep the same proportion to one another. The same relation can exist even when the numbers in it are changed. The universality of the relationship as such transcends its components."

Gadamer here identifies in proportionality one of the key features of Thurstone's and Rasch's measurement models, the capacity through which different items can provoke different responses from different people but still remain consistent with one another and provide comparable measures, as in computer adaptive testing (Lunz, Bergstrom, & Gershon 1994). Gadamer (1980, p. 149) also understands that "the real problem in the logos lies in its being the unity of an opinion composed of factors or items which are distinct from the opinion itself. Now, as we know, logos is a mathematical term that means 'proportion.'"

These comments are quite reminiscent of Thurstone (1928, p. 228): "If the scale is to be regarded as valid, the scale values of the statements should not be affected by the opinions of the people who help to construct it."

In other words, for measures to represent the logos of an object of discourse, the factors or items instrumental to that representation must remain in constant proportion to one another, so as to be separable and distinct from the opinion itself. Thurstone, writing 40 years before Gadamer's article first appeared in German, appears to have applied a principle of reason fundamental to science since Plato.

As is well known, Thurstone (1928, p. 228) characterizes checks on the extent to which scale values are or are not affected by the opinions of the people who help to calibrate the tool as a "crucial experimental test." This test is rarely employed by psychologists, who apparently find vulnerability to falsification more of a fault than a virtue (Michell 1990, p. 130). The test is, however, routinely implemented in Rasch measurement (Andrich 1978; Smith 2000).

It is remarkable then that Gadamer goes so far as to pinpoint the crux of what Andrich (2002) calls "resistance to the data-model relationship in Rasch's paradigm," saying that "the test which is to be applied in respect to the eidos [the logos of a particular idea] is a test of the immanent, internal coherence of all that is intrinsic to it. One should go no further until one is clear about what the assumption of the eidos means and what it does not mean. It should be noted that consequently the hypothesis is not to be tested against presumed empirical consequences, but conversely the empirical consequences are to be tested against the hypothesis, i.e., that from the start everything empirical or accidental which the eidos does not mean and imply is to be excluded from consideration. This means above all that the particular which participates in an eidos is of importance in an argument only in regard to that in which it may be said to participate, i.e., only in regard to its eidetic content." Gadamer (1980: 33-4; 1986: 101-2).

This passage conveys the essential importance of instrument calibration as the isolation of a particular thing, a variable or construct, that dominates a repeatably identifiable object of discourse. As was repeatedly stressed by Messick (for instance, 1975) in his work on construct validity, making measures inherently assumes that responses to questions embody a certain internal coherence, and so measures certainly should not be subjected to statistical comparisons until we are clear about what they mean and do not mean.

Unfortunately, IRT and classical test theory (CTT) begin from the position that the hypothetical model of what is being measured (referred to by Gadamer as the hypothesis of the eidos) is tested against the data (the presumed empirical consequences). In this paradigm, the model that best describes the data is taken as the basis for instrument calibration, even when that model explicitly (in the IRT case) or implicitly (in the CTT case) incorporates parameters that make it impossible to separate the particular factors or items involved in a unitary opinion from that opinion.

But as Gadamer says in the sentence immediately preceding the passage just quoted, "Such a procedure would be totally absurd in respect to a postulated eidos: that which constitutes being a horse could never be proved or disproved by a particular horse." It is common practice, however, in the implementation of IRT models with multiple item parameters, to decide that that which constitutes reading ability or moral development is proved or disproved by particular items or factors that are not distinct from the particular abilities of the persons measured.

Rasch models, in contrast, test empirical consequences against the hypothesized construct, holding, precisely in accord with Gadamer, that any test or survey question in particular is important only to the extent that it actually participates in and contributes to the generalizable measurement of the object of interest by being separable from it.

Ever since Kuhn's 1962 extension of the linguistic concept of the paradigm to the history of science, we have come to a fuller appreciation of the fact that "...reason has insight only into that which it produces after a plan of its own, and that it must not allow itself to be kept, as it were, in nature's leading-strings, but must itself show the way with principles of judgment based upon fixed laws, constraining nature to give answer to questions of reason's own determining. Accidental observations, made in obedience to no previously thought-out plan, can never be made to yield a necessary law, which alone reason is concerned to discover. Reason, holding in one hand its principles, according to which alone concordant appearances can be admitted as equivalent to laws, and in the other hand the experiment which it has devised in conformity with these principles, must approach nature in order to be taught by it. It must not, however, do so in the character of a pupil who listens to everything that the teacher chooses to say, but of an appointed judge who compels the witnesses to answer questions which he himself has formulated. ... It is thus that the study of nature has entered on the secure path of a science, after having for so many centuries been nothing but a process of merely random groping" (Kant 1965: 20-1).

As long as IRT and CTT dominate test- and survey-based measurement, we can expect nothing but continued random groping from the human sciences, since "the road from scientific law to scientific measurement can rarely be traveled in the reverse direction" (Kuhn 1977, p. 219). Rasch models specify the structure of scientific laws (Rasch 1960, p. 110-5) and so provide a framework in which reason can have insight through the projection of a plan of its own, showing the way with principles of judgment based on necessary and sufficient lawful relations.

For those seriously interested in pursuing this line of thought, I strongly recommend close and repeated reading of Heidegger's (1967, 1977) book, What is a thing? Fuller treatments of these ideas are taken up in my own recent work (Fisher 2003a, 2003b, 2003c, 2004).

William P. Fisher, Jr.

Andrich, D. A. (1978). Relationships between the Thurstone and Rasch approaches to item scaling. Applied Psychological Measurement, 2, 449-460.

Andrich, D. A. (2002). Understanding Rasch measurement: Understanding resistance to the data-model relationship in Rasch's paradigm: A reflection for the next generation. Journal of Applied Measurement, 3(3), 325-59.

Descartes, R. (1961). Rules for the direction of the mind. Indianapolis: Bobbs-Merrill.

Gadamer, H.-G. (1979). Historical transformations of reason. In T. F. Geraets (Ed.), Rationality today (pp. 3-14). Ottawa, Canada: University of Ottawa Press.

Gadamer, H.-G. (1980). Dialogue and dialectic: Eight hermeneutical studies on Plato (P. C. Smith, Trans.). New Haven: Yale University Press.

Gadamer, H.-G. (1986). The idea of the good in Platonic-Aristotelian philosophy (P. C. Smith, Trans.). New Haven: Yale University Press.

Gadamer, H.-G. (1989). Truth and method (J. Weinsheimer & D. G. Marshall, Trans.) (Rev. ed.). New York, New York: Crossroad (Original work published 1960).

Heidegger, M. (1967). What is a thing? (W. B. Barton, Jr. & V. Deutsch, Trans.). South Bend, Indiana: Regnery/Gateway.

Heidegger, M. (1977). Modern science, metaphysics, and mathematics (W. B. Barton, & V. Deutsch, Trans.). In D. F. Krell, (Ed.). Basic writings (pp. 243-282). New York, New York: Harper & Row. Rpt. from M. Heidegger, What is a Thing? South Bend, Indiana: Regnery/Gateway, pp. 66-108.

Kant, I. (1929/1965). Critique of pure reason (N. K. Smith, Trans.) (Unabridged). New York, New York: St. Martin's Press.

Kuhn, T. S. (1977). The function of measurement in modern physical science. In T. S. Kuhn, The essential tension: Selected studies in scientific tradition and change. Chicago, Illinois: University of Chicago Press (pp. 178-224). Reprinted from T. S. Kuhn, (1961), The function of measurement in modern physical science. Isis, 52(168), 161-193.

Lunz, M. E., Bergstrom, B. A., & Gershon, R. C. (1994). Computer adaptive testing. International Journal of Educational Research, 21(6), 623-634.

Messick, S. (1975, October). The standard problem: Meaning and values in measurement and evaluation. American Psychologist, 30, 955-966.

Michell, J. (1990). An introduction to the logic of psychological measurement. Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests (Reprint, with Foreword and Afterword by B. D. Wright, Chicago: University of Chicago Press, 1980). Copenhagen, Denmark: Danmarks Paedogogiske Institut.

Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (pp. 321-333). Berkeley, California: University of California Press.

Smith, R. M. (2000). Fit analysis in latent trait measurement models. Journal of Applied Measurement, 1(2), 199-218.

Thurstone, L. L. (1928). Attitudes can be measured. American Journal of Sociology, XXXIII, 529-544. Reprinted in L. L. Thurstone, The Measurement of Values. Midway Reprint Series. Chicago, Illinois: University of Chicago Press, 1959, pp. 215-233.

Plato's Separability Theorem, Fisher W.P.Jr. … Rasch Measurement Transactions, 2003, 17:3 p.939-941

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