Godfrey Thomson, Charles Spearman, and the mathematics of general intelligence

The early 20th century research on human intelligence prominently featured a disagreement between Charles Spearman and Godfrey Thomson regarding the interpretation of correlations among mental-test scores. Spearman argued that the pattern of positive correlations commonly observed among diverse cognitive tests—the positive manifold—arises because performance on all tests reflects a single general factor of intelligence, g, with other test variance being unique to each test. Thomson responded that the mathematical structure of the correlations does not logically entail the existence of a single underlying general factor—it might, but he thought other explanations were possible. The present paper revisits Thomson's mathematical critique of Spearman's model in two new ways: it focuses on a substantial, important, and often-ignored paper by Thomson in 1935; and it separates the mathematical arguments about the correlations among cognitive tests from the neurobiological speculations that decorated and distracted from them. Using modern notation, we restate Spearman's single-factor formulation and show how it implies a hierarchical correlational structure characterised by near-to-zero “tetrad differences”. We then present Thomson's alternative formulation, in which test scores arise from combinations of numerous underlying components, and demonstrate that this model generates the same correlation pattern. We further note that infinitely many algebraic transformations of Spearman's model can result in the same correlational structure. Therefore, the presence of hierarchically-ordered, positive correlations might imply, but does not uniquely identify, a single general factor. The paper clarifies the mathematical basis of the Spearman–Thomson debate and distinguishes it from psychological and biological interpretations.