User:IssaRice/Implementation details in math

ideas similar to software engineering's "implementation details" come up quite often in math, when we are constructing/defining certain objects:

• constructing number systems
• thinking of relations as subset of the Cartesian product
• thinking of functions as a particular kind of relation
• thinking of events in probability theory as sets
• thinking about the sample space in probability

It seems to me that these implementation details are not really needed after some point, and yet they are quite helpful pedagogically.