soft question - What does a "convention" mean in mathematics?

We all know that $0!=1$, the degree of the zero polynomial equals $-\infty$, the interval$[a,a)=(a,a]=(a,a)=\emptyset$ ... and so on, are conventions in mathematics. So is a convention something that we can't prove with mathematical logic, or is it just intuitions, or something
that mathematicians agree about?
Are they the same as axioms? What does "convention" mean in mathematics?
And is $i^2 = -1$ a convention? If not how can we prove existence of such number?

Answer

To answer the question in the title, I would say: 'convention' in mathematics means exactly the same as in ordinary English.

As for your examples: $0!:=1$ and $[a,a):=\emptyset$ are definitions. It is a convention not to use a different definition, or to leave it undefined. Of course in this sense, every definition is a convention.

It think that informally, one says a certain definition (such as the two above) is '(just) convention', to mean that they are 'extreme' or 'degenerate' cases, and leaving them undefined would still make the theory go through, but it is more convenient to define them anyway (for example to prevent having to exclude this extreme case in statement of theorems). For example, I think you could get by not defining $[a,a)$ or $[a,b]$ for $ba$ which could be tiresome.