Greek theoretical mathematics emerges among sixth-century Ionians from a background of professional practitioners, concerned chiefly with arithmetical operations. Its characteristical features (impersonalization, standardization, diagrams) develop as part of an elitist play of distinction at fifth- and fourth-century Athens, mainly to ascertain the tradition of that knowledge without adequate institutions. Sophists challenge the mathematicians' practices, philosophers adopt the mathematicians' knowledge as a model of truth, but the mathematicians themselves remain autonomous. Even hellenistic mathematics, semi-institutionalized at royal courts, is little more than a private affair of a narrow circle of intellectuals. All this time the practitioners' traditions persist basically unaltered and constantly present the theoreticians with a social and epistemic background to differentiate "their" mathematics from. Finally, these two branches of Greek mathematics, a practical and a theoretical one, are describable as reciprocally systematized forms of knowledge.