QuoteFermion:Axioms depend on definitions. Axioms are not beliefs but are postulated. Right, axioms are nothing more than the antecedent "p" in a conditional "p ----> q", where "p" and "q" themselves may be compound statements (or propositions). And we may think of the consequent "q" as theorems (or the results). There is no need for believing the axioms as true (or false). In fact, one of the easiest and most utterly useless of all deductive systems is one in which the axioms are patently contradictory, within which any result can be derived. A mathematician (anyone in fact) can do both Euclidean and Non-Euclidean geometry without ever worrying as to which of the two has the believable (or true) set of axioms. This is why mathematics is independent of reality of the world, existence! But, axioms do not at all depend on definitions. Actually one can do math without any definition. The device of definition in math serves the same purpose as it does in ordinary speech; namely, to make communication more manageable. In an axiomatic-deductive system, there are always certain elementary or primitive terms that will forever remain not defined. For example, in euclidean geometry we take the 'notions' of "set", "point", "line" as undefined. Using these notions, we formulate the axioms as stating a relationship between these primitive terms. For example, we have the following axiom: Given any two distinct points, there exists a unique line containing these two points. So far we have not defined anything. Now, let's say we were interested in a certain subset of a line. The subset consisting of all points of the line that are 'between' two given points of the line. Now, in all our future consideration of such a subset, either we can restate our idea expressed in those many words, or we can just agree to give a special symbol or name to the idea expressed by all those words. We commonly use the name "line segment" to define (or symbolize) the idea of a subset consisting of all the points (of the line) between the two given points. So, by defining we never create any new knowledge. In fact some of the most amazing definitions apply to certain things that don't even exist! But since the idea of a (hitherto unrealized) impossible thing does exist, one can always define the idea by the choice of an appropriate word (or symbol).
Last edited by dunrewpp
on March 19th, 2010, 11:00 pm, edited 1 time in total.