The word "convention" is used in various ways, and you might be talking about various sets of people; but as we use the term in math, if the "people" you refer to are the mathematical community as a whole (rather than, say, you and me alone), then they mean essentially the same thing:Are they the same thing? Are there differences between these two?
I'm sorry, but I can't make out what you're trying to say here...?I was thinking that axioms has the convention that they can't aasume to be proven
I'm sorry, but on what basis have you concluded that non-Euclidean geometry is "debatable"? It's various forms are geometries, just like planar or spherical Euclidean geometries are geometries. They just start with different assumptions.but I was thinking that something that people can debate on it like using non-Euclidan geometry instead geometry.
In my experience, a "convention" is more like an arbitrary rule (such as on which side of the street you drive in a given country) or an accepted practice or process (like using BODMAS to remember your order of operations) or a nicety (such as asking a person "How are you?" when you're really just starting a conversation).So what people agree to use (different geometry) against it is not proven and everybody can except it - a Convetion.