Area of math that started from axioms

Question

I read somewhere that most areas of math don't start out with axioms. They become formulated later on.

Are there any areas of math, in which that area was first developed and expanded from axioms?

Answer 1

The way things usually work in math is that examples come first, and axioms second. Of course we prove things formally by starting with axioms. But the axioms themselves are found by considering a class of examples and trying to isolate the relevant properties.

So, although it would be possible in principle to start a new area of mathematics by arbitrarily choosing a set of axioms, that is not the way that mathematical research is actually done.

Answer 2

Here is what happens when an area is first developed and expanded from axioms.