Some Reflections on Numerals in Sumerian towards a History of Mathematical Speculation

Abstract

THE SUMERIAN NUMERALS HAVE MORE THAN ONCE since Samuel Noah Kramer's undergraduate days been the object of discussion, but still there is much that is puzzling about them. Unfortunately, the recent, comprehensive treatment of the subject by M. Powell became accessible to me only when this paper was practically finished.2 My study was induced by D. 0. Edzard's recent paper, where he treats the Sumerian numerals as listed in a school tablet from Ebla, and supplies much material upon which to reflect.3 Apparently, all that can be gleaned from the Sumerian material itself has by now been weighed and summed up, and since Sumerian has no known related languages, comparative historical research is impossible. Therefore I shall try to put the data in a typological context, comparing Sumerian numerals to numerals in other archaic languages. In order to escape subjectivity in using the latter term, I shall define as 'archaic' any language which, on the lexical level, has no or only poorly developed means of expressing abstract ideas, and on the grammatical level, is based on the opposition 'action vs. state' (or 'transitive action' vs. 'Intransitive action and state'). Such an opposition implies (a) that the subject of action (or of transitive action) is expressed by a specific oblique (ergative) case, while the subject of state, including the state resulting from an action, is expressed by the direct or absolute (usually zero) case, and (b) that a grammatical category of the direct object as distinct from the subject of a state does not exist in the syntax, nor that of the accusative case in the morphology.4 In an archaic language there are no adequate means, either lexical or grammatical, to express such abstract ideas as 'time', space', 'subject', 'object', 'cause', 'beauty', 'liberty', 'invention', 'multiplication', 'division' and many others, some of which appear to us elemental, as, e.g., the distinction between 'darkness', 'calamity', 'illness', and 'pain', etc., or between 'good', 'enjoyable', 'kind', 'happy', 'useful', 'lucky', etc. However, human thought is impossible without inductive thinking, i.e., thought which proceeds from particular facts to a generalization. In the absence of means to express general ideas, one resorts to generalization by tropes (metaphors and metonymies). All myths are actually generalizations of the empirical facts of life through tropes, and, like all tropes, they are pregnant with man's emotional attitudes to the facts. The development of lexical and grammatical means to express general ideas opens the way, on the one hand, for scientific thought, which has as its goal a nonemotional cognition of objects, and on the other, for specifically artistic thought, whose goal is the emotional cognition of man's attitude toward and relation to objects.