G
Guest
Guest
DO mathematicians believe in Axiom like some people believe in their GOD ?
- Joined
- Apr 12, 2005
- Messages
- 11,339
I think it a reasonable question. Generally, the answer is an emphatic "NO". Axioms are simply fundamental assumptions. There is nothing sacred about them.
Having said that, rejecting an axiom oftentimes irritates those who are really not getting what they have been studying. Wander through a NonEuclidean Geometry class sometime. I suspect you will almost always find at least one student struggling mightily with releasing an axiom or two.
Having said that, rejecting an axiom oftentimes irritates those who are really not getting what they have been studying. Wander through a NonEuclidean Geometry class sometime. I suspect you will almost always find at least one student struggling mightily with releasing an axiom or two.
HallsofIvy
Elite Member
- Joined
- Jan 27, 2012
- Messages
- 7,763
More correctly, "axioms" define which mathematical system we are talking about. In Euclidean geometry we use the axiom (or "postulate"- Euclid used 'axiom' for what were basically arithmetic facts, 'postulate' for geometric statements. Modern mathematics doesn't really make a distinction.) "through a given point not on a given line there exist exactly one line parallel to the given line". In hyperbolic geometry we use "through a given point not on a given line there exist an infinite number of lines parallel to the given line". And in elliptic geometry we use "through a given point not on a given line there exist no line parallel to the given line."
There is no "assumption" that any given statement is true or false- only whether we want to use it in whichever mathematical structure we are working with.
There is no "assumption" that any given statement is true or false- only whether we want to use it in whichever mathematical structure we are working with.
Last edited: