Gradience as a cognitive principle for evaluating numerical notations

More than 100 historically, archaeologically and ethnographically attested numerical notations have been used over the past 5,000 y; however, because most of those systems are no longer used, experimental investigation is challenging. Prior research often assumes, rather than demonstrates, the inferiority of notations like Roman numerals. Gradience is a principle whereby the length of numeral phrases correlates with the magnitude of the numbers being represented. In general, but inconsistently, larger numbers require more signs, just as larger sets of abstract dot patterns occupy more space. This study compares the gradience of 13 numerical notations varying in phylogenetic family, linguistic family, and structural properties, using two indices, inversion (how often N + 1 requires fewer signs than N) and jitter (the mean length difference between successive numbers). Place value systems are highly gradient; i.e., their length indexes numerical magnitude more accurately than other systems. The relationship between the structural properties of notations and the two indices of jitter and inversion is complex. Next, a cultural-evolutionary analysis compares the older additive Roman numerals to the eventually predominant subtractive variant (e.g., XIX = 19). Subtractive Roman numerals are more concise and have a lower jitter than additive ones, but at the cost of a significantly higher inversion ratio. This analysis permits the evaluation of cultural-evolutionary hypotheses grounded in the representational properties of attested notations, even when cognitive tests are not feasible. Several avenues for future experimental investigation using attested or constructed notations of varying properties are proposed.