Abstract

ABSTRACT In Greek Mathematical Thought and the Origin of Algebra, Jacob Klein contrasts ancient Greek philosophy's direct engagement with things through arithmetic with the ancient science of numeric calculation, logistic. By chronicling the later development of logistic, by means of increasing symbolization, ultimately into algebra, he argues that logistic has come to displace arithmetic and, thereby, to submerge the ontological issues at the center of Greek thought. This article argues, first, that Klein's target is Ernst Cassirer's notion of number as a “symbolic function”—what Cassirer would later call “symbolic form.” In Substance and Function Cassirer argues that number constitutes a conceptual structure progressively extended by thought so as to include the results of different sorts of calculation within itself and, thereby, to be a closed system. Drawing upon Dedekind, Cassirer claims that no individual number can be understood apart from the entire conceptual structure. This article argues, second, that Cassirer recognizes, as it were, a different set of things, namely, relations, and that it is possible to engage these “things” directly by recognizing the ontological issues they raise.