chapter  XIX
Hypothetical-Deductive Systems, Non-Contra- dictoriness, Antinomies
Pages 10

All these together brought about what Enriques called the reform of contemporary logic, by which we have come to consider every mathematical theory as a hypothetical-deductive system.

This term, introduced by M. Pieri (1860-1913) (2), indicates a system of propositions consisting of postulates and theorems : the postulates (which from the purely logical point of view are arbitrary) define implicitly the primitive concepts introduced in the theory without explicit definitions ; but these concepts are defined up to a point, because they can still be interpreted concretely in several ways, for instance, physico-mathematically, geometrically, purely analytically; the theorems are obtained from the postulates by applying the laws of formal logic without an appeal to intuition. Bertrand Russell has

written paradoxically : 'Mathematics is the science in which we do not know what we are talking about nor whether what we are saying is true or false. '