Unilinear ranking of pupils has no place, even in Thurstone’s world of just a few PMA’s. The essence of each child becomes his individuality, Thurstone wrote (1935, p. 53):

  Even if each individual can be described in terms of a limited number of independent reference abilities, it is still possible for every person to be different from every other person in the world. Each person might be described in terms of his standard scores in a limited number of independent abilities. The number of permutations of these scores would probably be sufficient to guarantee the retention of individualities.

  From the midst of an economic depression that reduced many of its intellectual elite to poverty, an America with egalitarian ideals (however rarely practiced) challenged Britain’s traditional equation of social class with innate worth. Spearman’s g had been rotated away, and general mental worth evaporated with it.

  One could read the debate between Burt and Thurstone as a mathematical argument about the location of factor axes. This would be as myopic as interpreting the struggle between Galileo and the Church as an argument between two mathematically equivalent schemes for describing planetary motion. Burt certainly understood this larger context when he defended the 11+ examination against Thurstone’s assault:

  In educational practice the rash assumption that the general factor has at length been demolished has done much to sanction the impracticable idea that, in classifying children according to their varying capabilities, we need no longer consider their degree of general ability, and have only to allot them to schools of different types according to their special aptitudes; in short, that the examination at 11 plus can best be run on the principle of the caucus-race in Wonderland, where everybody wins and each get some kind of prize (1955, p. 165).

  Thurstone, for his part, lobbied hard, producing arguments (and alternate tests) to support his belief that children should not be judged by a single number. He wished, instead, to assess each person as an individual with strengths and weaknesses according to his scores on an array of PMA’s (as evidence of his success in altering the practice of testing in the United States, see Guilford, 1959, and Tuddenham, 1962, p. 515).

  Instead of attempting to describe each individual’s mental endowment by a single index such as a mental age or an intelligence quotient, it is preferable to describe him in terms of a profile of all the primary factors which are known to be significant.… If anyone insists on having a single index such as an I.Q., it can be obtained by taking an average of all the known abilities. But such an index tends so to blur the description of each man that his mental assets and limitations are buried in the single index (1946, p. 110).

  Two pages later, Thurstone explicitly links his abstract theory of intelligence with preferred social views.

  This work is consistent not only with the scientific object of identifying the distinguishable mental functions but it seems to be consistent also with the desire to differentiate our treatment of people by recognizing every person in terms of the mental and physical assets which make him unique as an individual (1946, p. 112).

  Thurstone produced his fundamental reconstruction without attacking either of the deeper assumptions that had motivated Spearman and Burt—reification and hereditarianism. He worked within established traditions of argument in factor analysis, and reconstructed results and their meaning without altering the premises.

  Thurstone never doubted that his PMA’s were entities with identifiable causes (see his early work of 1924, pp. 146–147, for the seeds of commitment to reifying abstract concepts—gregariousness in this case—as things within us). He even suspected that his mathematical methods would identify attributes of mind before biology attained the tools to verify them: “It is quite likely that the primary mental abilities will be fairly well isolated by the factorial methods before they are verified by the methods of neurology or genetics. Eventually the results of the several methods of investigating the same phenomena must agree” (1938, p. 2).

  The vectors of mind are real, but their causes may be complex and multifarious. Thurstone admitted a strong potential influence for environment, but he emphasized inborn biology:

  Some of the factors may turn out to be defined by endocrinological effects. Others may be defined by biochemical or biophysical parameters of the body fluids or of the central nervous system. Other factors may be defined by neurological or vascular relations in some anatomical locus; still others may involve parameters in the dynamics of the autonomic nervous system; still others may be defined in terms of experience and schooling (1947, p. 57).

  Thurstone attacked the environmentalist school, citing evidence from studies of identical twins for the inheritance of PMA’s. He also claimed that training would usually enhance innate differences, even while raising the accomplishments of both poorly and well-endowed children:

  Inheritance plays an important part in determining mental performance. It is my own conviction that the arguments of the environmentalists are too much based on sentimentalism. They are often even fanatic on this subject. If the facts support the genetic interpretation, then the accusation of being undemocratic must not be hurled at the biologists. If anyone is undemocratic on this issue, it must be Mother Nature. To the question whether the mental abilities can be trained, the affirmative answer seems to be the only one that makes sense. On the other hand, if two boys who differ markedly in visualizing ability, for example, are given the same amount of training with this type of thinking, I am afraid that they will differ even more at the end of the training than they did at the start (1946, p. 111).

  As I have emphasized throughout this book, no simple equation can be made between social preference and biological commitment. We can tell no cardboard tale of hereditarian baddies relegating whole races, classes, and sexes to permanent biological inferiority—or of environmentalist goodies extolling the irreducible worth of all human beings. Other biases must be factored (pardon the vernacular usage) into a complex equation. Hereditarianism becomes an instrument for assigning groups to inferiority only when combined with a belief in ranking and differential worth. Burt united both views in his hereditarian synthesis. Thurstone exceeded Burt in his commitment to a naïive form of reification, and he did not oppose hereditarian claims (though he certainly never pursued them with the single-minded vigor of a Burt). But he chose not to rank and weigh on a single scale of general merit, and his destruction of Burt’s primary instrument of ranking—Spearman’s g—altered the history of mental testing.

  Spearman and Burt react

  When Thurstone dispersed g as an illusion, Spearman was still alive and pugnacious as ever, while Burt was at the height of his powers and influence. Spearman, who had deftly defended g for thirty years by incorporating critics within his flexible system, realized that Thurstone could not be so accommodated:

  Hitherto all such attacks on it [g] appear to have eventually weakened into mere attempts to explain it more simply. Now, however, there has arisen a very different crisis; in a recent study, nothing has been found to explain; the general factor has just vanished. Moreover, the said study is no ordinary one. Alike for eminence of the author, for judiciousness of plan, and for comprehensiveness of scope, it would be hard to find any match for the very recent work on Primary Mental Abilities by L. L. Thurstone (Spearman, 1939, p. 78).

  Spearman admitted that g, as an average among tests, could vary in position from battery to battery. But he held that its wandering was minor in scope, and that it always pointed in the same general direction, determined by the pervasive positive correlation between tests. Thurstone had not eliminated g; he had merely obscured it by a mathematical dodge, distributing it by bits and pieces among a set of group factors: “The new operation consisted essentially in scattering g among such numerous group factors, that the fragment assigned to each separately became too small to be noticeable” (1939, p. 14).

  Spearman then turned Thurstone’s favorite argument against him. As a convinced reifier, Thurstone believed th
at PMA’s were “out there” in fixed positions within a factorial space. He argued that Spearman and Burt’s factors were not “real” because they varied in number and position among different batteries of tests. Spearman retorted that Thurstone’s PMA’s were also artifacts of chosen tests, not invariant vectors of mind. A PMA could be created simply by constructing a series of redundant tests that would measure the same thing several times, and establish a tight cluster of vectors. Similarly, any PMA could be dispersed by reducing or eliminating the tests that measure it. PMA’s are not invariant locations present before anyone ever invented tests to identify them; they are products of the tests themselves:

  We are led to the view that group factors, far from constituting a small number of sharply cut “primary” abilities, are endless in number, indefinitely varying in scope, and even unstable in existence. Any constitutent of ability can become a group factor. Any can cease being so (1939, p. 15).

  Spearman had reason to complain. Two years later, for example, Thurstone found a new PMA that he could not interpret (in Thurstone and Thurstone, 1941). He called it Xj and identified it by strong correlations between three tests that involved the counting of dots. He even admitted that he would have missed Xi entirely, had his battery included but one test of dotting:

  All these tests have a factor in common; but since the three dot-counting tests are practically isolated from the rest of the battery and without any saturation on the number factor, we have very little to suggest the nature of the factor. It is, no doubt, the sort of function that would ordinarily be lost in the specific variance of the tests if only one of these dot-counting tests had been included in the battery (Thurstone and Thurstone, 1941, pp. 23–24).

  Thurstone’s attachment to reification blinded him to an obvious alternative. He assumed that X1 really existed and that he had previously missed it by never including enough tests for its recognition. But suppose that X1 is a creation of the tests, now “discovered” only because three redundant measures yield a cluster of vectors (and a potential PMA), whereas one different test can only be viewed as an oddball.

  There is a general flaw in Thurstone’s argument that PMA’s are not test-dependent, and that the same factors will appear in any properly constituted battery. Thurstone claimed that an individual test would always record the same PMA’s only in simple structures that are “complete and overdetermined” (1947, p. 363)—in other words, only when all the vectors of mind have been properly identified and situated. Indeed, if there really are only a few vectors of mind, and if we can know when all have been identified, then any additional test must fall into its proper and unchanging position within the invariant simple structure. But there may be no such thing as an “overdetermined” simple structure, in which all possible factor axes have been discovered. Perhaps the factor axes are not fixed in number, but subject to unlimited increase as new tests are added. Perhaps they are truly test-dependent, and not real underlying entities at all. The very fact that estimates for the number of primary abilities have ranged from Thurstone’s 7 or so to Guilford’s 120 or more indicates that vectors of mind may be figments of mind.

  If Spearman attacked Thurstone by supporting his beloved g, then Burt parried by defending a subject equally close to his heart—the identification of group factors by clusters of positive and negative projections on bipolar axes. Thurstone had attacked Spearman and Burt by agreeing that factors must be reified, but disparaging the English method for doing so. He dismissed Spearman’s g as too variable in position, and rejected Burt’s bipolar factors because “negative abilities” cannot exist. Burt replied, quite properly, that Thurstone was too unsubtle a reifier. Factors are not material objects in the head, but principles of classification that order reality. (Burt often argued the contrary position as well—see pp. 318–322.) Classification proceeds by logical dichotomy and antithesis (Burt, 1939). Negative projections do not imply that a person has less than zero of a definite thing. They only record a relative contrast between two abstract qualities of thought. More of something usually goes with less of another—administrative work and scholarly productivity, for example.

  As their trump card, both Spearman and Burt argued that Thurstone had not produced a cogent revision of their reality, but only an alternative mathematics for the same data.

  We may, of course, invent methods of factorial research that will always yield a factor-pattern showing some degree of “hierarchical” formation of (if we prefer) what is sometimes called “simple structure.” But again the results will mean little or nothing: using the former, we could almost always demonstrate that a general factor exists; using the latter, we could almost always demonstrate, even with the same set of data, that it does not exist (Burt, 1940, pp. 27–28).

  But didn’t Burt and Spearman understand that this very defense constituted their own undoing as well as Thurstone’s? They were right, undeniably right. Thurstone had not proven an alternate reality. He had begun from different assumptions about the structure of mind and invented a mathematical scheme more in accord with his preferences. But the same criticism applies with equal force to Spearman and Burt. They too had started with an assumption about the nature of intelligence and had devised a mathematical system to buttress it. If the same data can be fit into two such different mathematical schemes, how can we say with assurance that one represents reality and the other a diversionary tinkering? Perhaps both views of reality are wrong, and their mutual failure lies in their common error: a shared belief in the reification of factors.

  Copernicus was right, even though acceptable tables of planetary positions can be generated from Ptolemy’s system. Burt and Spearman might be right even though Thurstone’s mathematics treats the same data with equal facility. To vindicate either view, some legitimate appeal must be made outside the abstract mathematics itself. In this case, some biological grounding must be discovered. If biochemists had ever found Spearman’s cerebral energy, if neurologists had ever mapped Thurstone’s PMA’s to definite areas of the cerebral cortex, then the basis for a preference might have been established. All combatants made appeals to biology and advanced tenuous claims, but no concrete tie has even been confirmed between any neurological object and a factor axis.

  We are left only with the mathematics, and therefore cannot validate either system. Both are plagued with the conceptual error of reification. Factor analysis is a fine descriptive tool; I do not think that it will uncover the elusive (and illusory) factors, or vectors, of mind. Thurstone dethroned g not by being right with his alternate system, but by being equally wrong—and thus exposing the methodological errors of the entire enterprise.*

  Oblique axes and second-order g

  Since Thurstone pioneered the geometrical representation of tests as vectors, it is surprising that he didn’t immediately grasp a technical deficiency in his analysis. If tests are positively correlated, then all vectors must form a set in which no two are separated by an angle of more than 90° (for a right angle implies a correlation coefficient of zero). Thurstone wished to put his simple structure axes as near as possible to clusters within the total set of vectors. Yet he insisted that axes be perpendicular to each other. This criterion guarantees that axes cannot lie really close to clusters of vectors—as Fig. 6.11 indicates. For the maximal separation of vectors is less than 90°, and any two axes, forced to be perpendicular, must therefore lie outside the clusters themselves. Why not abandon this criterion, let the axes themselves be correlated (separated by an angle of less than 90°), and permit them to lie right within the clusters of vectors?

  Perpendicular axes have a great conceptual advantage. They are mathematically independent (uncorrelated). If one wishes to identify factor axes as “primary mental abilities,” perhaps they had best be uncorrelated—for if factor axes are themselves correlated, then doesn’t the cause of that correlation become more “primary” than the factors themselves? But correlated axes also have a different kind of conceptual advantage: they can be placed
nearer to clusters of vectors that may represent “mental abilities.” You can’t have it both ways for sets of vectors drawn from a matrix of positive correlation coefficients: factors may be independent and only close to clusters, or correlated and within clusters. (Neither system is “better”; each has its advantages in certain circumstances. Correlated and uncorrelated axes are both still used, and the argument continues, even in these days of computerized sophistication in factor analysis.)

  Thurstone invented rotated axes and simple structure in the early 1930s. In the late 1930s he began to experiment with so-called oblique simple structures, or systems of correlated axes. (Uncorrelated axes are called “orthogonal” or mutually perpendicular; correlated axes are “oblique” because the angle between them is less than 90°.) Just as several methods may be used for determining orthogonal simple structure, oblique axes can be calculated in a variety of ways, though the object is always to place axes within clusters of vectors. In one relatively simple method, shown in Fig. 6.11, actual vectors occupying extreme positions within the total set are used as factor axes. Note, in contrasting Figs. 6.7 and 6.11, how che factor axes for verbal and mathematical skills have moved from outside the actual clusters (in the orthogonal solution) to the clusters themselves (in the oblique solution).

  Most factor-analysts work upon the assumption that correlations may have causes and that factor axes may help us to identify them. If the factor axes are themselves correlated, why not apply the same argument and ask whether this correlation reflects some higher or more basic cause? The oblique axes of a simple structure for mental tests are usually positively correlated (as in Fig. 6.11). May not the cause of this correlation be identified with Spearman’s g? Is the old general factor ineluctable after all?