I disagree maths “should be” done differently. I have a strong feeling the way stuff is defined usually nowadays has a property of being maximally easy to use. We don’t really need the definitions to look exactly like the intuition we had to invent them as long as the resulting objects behave exactly the same, and the less intuitive definitions are easier to use in proofs. For example, defining all powers directly as the Taylor series of e^x makes defining complex and matrix exponentials much easier / possible at all, and ad hoc proof this coincides with the naive version is simple. Also simplifies checking well-definedness a lot. Many more such examples.
I disagree maths “should be” done differently. I have a strong feeling the way stuff is defined usually nowadays has a property of being maximally easy to use. We don’t really need the definitions to look exactly like the intuition we had to invent them as long as the resulting objects behave exactly the same, and the less intuitive definitions are easier to use in proofs. For example, defining all powers directly as the Taylor series of e^x makes defining complex and matrix exponentials much easier / possible at all, and ad hoc proof this coincides with the naive version is simple. Also simplifies checking well-definedness a lot. Many more such examples.