There are some foundational questions one can ask in calculus, and they lead
to different kinds of math and different approaches to calculus.

They involve making rigorous different kinds of intuition, and are valuable
(even if not adopted) for the perspectives they offer.

Note that they all involve their own technicalities, which advocates often
overlook (come now, a little bit of logic is a small price to pay to develop
infintesimals, right?).
This is not to say that standard calculus development doesn't involve
technicalities of a different sort, just that any approach requires -some-
technicality.

Further, in use (by engineers, say), they just do calc formally,
and prove it/understand it intuitively.
Formality is just necessary for proofs
...but of course also to avoid contradictions
and for deeper understanding.

The nomenclature comes from model theory:
(what I now say is vague and incorrect)
if you have a theory with a complete model,
then this is called the "standard model",
but other (non-complete) models are called "non-standard models"
