I heavily disagree; axioms are never a good place to
start, nor, historically, they ever
were a starting place.
Working mathematicians do a lot of mathematics and constructions before stopping to look and say: "OK, we can distill this rich theory down to this small set of axioms".
Geometry has been quite developed before Euclid's axioms, and number theory took its sweet millenia before Peano's axioms were developed.
On a more modern side, a lot of topology has been produced before people decided to write down a list of axioms for homology, which all the different homology constructions satisfied.
The point is that understanding almost always precedes formal reasoning, language follows ideas, and rigid systems arise on a fertile soil of messy experiments - mental or physical.
If you look at the history of science and mathematics, nothing was ever built up from the formulas and axioms (criminally contrary to the way these subjects are taught).
There is an underappreciated beauty in our ability to argue about, reason with, and make use of concepts that we haven't even really defined. Everyone knew Newton's Calculus was full of holes, and people used it for centuries before coming up with solid definitions of limits, derivatives and integrals - the most fundamental concepts of the subject!
And on that note, people used real numbers for a long time before having the formal machinery to define them. Zeno didn't have the axioms to resolve his paradox. Didn't stop anyone from using these weird objects in reasoning and practice.
----------------------------------------------------
TL;DR: never start with axioms, end with them.