Not entirely fair. While there is a lot of choice about how to define stuff around the edges of mathematics, after the axiomatic choices have been made, there'

Not entirely fair. While there is a lot of choice about how to define stuff around the edges of mathematics, after the axiomatic choices have been made, there's quite a lot to discover in their structure. Particularly if the relevant axioms relate to something outside of mathematics (the examples are too many to even scratch but start out thinking of the definitions of natural and rational numbers) then you really are discovering things in the same way as a physicist - in fact, this is how many physicists go about discovering things, for better or worse. In most situations it makes sense to define exponentiation as repeated multiplication, and a^0 as the absence of multiplication by a, hence a^0 = 1 as the multiplicative identity. I wouldn't introduce the idea of anything else to a student unless they specifically asked me about one of the problems which can arise in choosing 0^0 = 1.