Mathematically speaking, what is an ‘Axiom’?
In: 621
One example of an axiom: 2 parallel lines will never meet.
And you have a bunch of axioms like these: given two points, there is exactly one line going through those 2 points. They all sounded like “duh of course it’s true” but it is useful to formalize them even if you can picture it’s truthfulness in your head. Once you formalize them, mathematicians can start building theorems using those axioms, and build more theorems on top of those first layer theorems, and so on.
But the second reason why it’s important to formalize them is because sometimes it’s not true. Take the first one I mentioned for example: 2 parallel lines will never meet. That may be true in Euclidean Geometry. But when you’re talking about Projective Geometry, the axioms include one additional point called “point at infinity,” and the axiom is modified as: 2 parallel lines meet in exactly 1 point, that is point at infinity. The list of theorems that can be derived from there are slightly different from Euclidean Geometry, although they smell the same.
Latest Answers