The internal representation of numbers

A principle of second-order isomorphism asserts that the functional similarities among internal representations parallel the structural similarities among the external objects to which those representations correspond. In accordance with this principle, subjects were asked to rate the similarities of all pairs of the numbers 0 through 9 in each of 24 conditions distinguished by the forms into which they were to be mentally transformed and then judged-including the visual forms of rows of dots and Arabic numerals, the auditory form of spoken English names, and the amodal form of the abstract concepts of the integers themselves. Similarity ratings were found to depend entirely upon the form in which the numbers were to be judged and not at all upon the form in which they were actually presented. Regardless of the form presented, multidimensional scaling solutions for each of the 24 conditions were readily interpretable in terms of structural properties of the form judged-curvature and closure for visual numerals; number of syllables, initial consonant, and vowel phoneme for auditory names; and numerical magnitude, odd-evenness, etc., for abstract concepts. Reanalyses of chronometric data are presented to illustrate how a fuller understanding of human cognitive processes may be attainable by using such structural models of internal representation in conjunction with studies of overt confusion and reaction time.