Hilbert , David ( 1967 ) Paul
نویسندگان
چکیده
axiomatics. A main feature of Hilbert’s axiomatization of geometry is that the axiomatic method is presented and practiced in the spirit of the abstract conception of axiomatics that arose at the end of the nineteenth century and which has been generally adopted in modern mathematics. It consists in abstracting from the intuitive meaning of the terms for the kinds of primitive objects (individuals) and for the fundamental relations and in understanding the assertions (theorems) of the axiomatized theory in a hypothetical sense, that is, as holding true for any interpretation or determination of the kinds of individuals and of the fundamental relations for which the axioms are satisfied. Thus, an axiom system is regarded not as a system of statements about a subject matter but as a system of conditions for what might be called a relational structure. Such a relational structure is taken as the immediate object of the axiomatic theory; its application to a kind of intuitive object or to a domain of natural science is to be made by means of an interpretation of the individuals and relations in accordance with which the axioms are found to be satisfied. This conception of axiomatics, of which Hilbert was one of the first advocates (and certainly the most influential), has its roots in Euclid’s Elements, in which logical reasoning on the basis of axioms is used not merely as a means of assisting intuition in the study of spatial figures; rather, logical dependencies are considered for their own sake, and it is insisted that in reasoning we should rely only on those properties of a figure that either are explicitly assumed or follow logically from the assumptions and axioms. This program was not strictly adhered to in all parts of the Elements, nor could it have been, for its system of axioms was not sufficient for the purpose. The
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