Читать книгу The Philosopher's Toolkit - Julian Baggini, Julian Baggini - Страница 68
Second type of axiom
ОглавлениеAnother type of axiom is also true by definition, but in a slightly more interesting way. Many regions of mathematics and geometry rest on their axioms, and it’s only by accepting these basic axioms that more complex proofs can be constructed within those regions. You might call these propositions ‘primitive’ sentences within the system (7.7). For example, it is an axiom of Euclidean geometry that the shortest distance between any two points is a straight line. But while axioms like these are vital in geometry and mathematics, they merely stipulate what is true within the particular system of geometry or mathematics to which they belong. Their truth is guaranteed, but only in a limited way – that is, only within the context in which they’re defined. Used in this way, axioms’ acceptability rises or falls with the acceptability of the theoretical system as a whole.