Method of Analysis: A Paradigm of Mathematical Reasoning?

Abstract

The ancient Greek method of analysis has a rational reconstruction in the form of the tableau method of logical proof. This reconstruction shows that the format of analysis was largely determined by the requirement that proofs could be formulated by reference to geometrical figures. In problematic analysis, it has to be assumed not only that the theorem to be proved is true, but also that it is known. This means using epistemic logic, where instantiations of variables are typically allowed only with respect to known objects. This requirement explains the preoccupation of Greek geometers with questions as to which geometrical objects are ‘given’, that is, known or ‘data’, as in the title of Euclid's eponymous book. In problematic analysis, constructions had to rely on objects that are known only hypothetically. This seems strange unless one relies on a robust idea of ‘unknown’ objects in the same sense as the unknowns of algebra. The Greeks did not have such a concept, which made their grasp of the analytic method shaky.