  This article was motivated by the discovery of a potential new foundation for
mainstream mathematics. The goals are to clarify the relationships between
primitives, foundations, and deductive practice; to understand how to determine
what is, or isn't, a foundation; and get clues as to how a foundation can be
optimized for effective human use.
  For this we turn to history and professional practice of the subject. We have
no asperations to Philosophy.
  The first section gives a short abstract discussion, focusing on the
significance of consistency. The next briefly describes foundations, explicit
and implicit, at a few key periods in mathematical history. We see, for
example, that at the primitive level human intuitions are essential, but can be
problematic. We also see that traditional axiomatic set theories,
Zermillo-Fraenkel-Choice (ZFC) in particular, are not quite consistent with
mainstream practice. The final section sketches the proposed new foundation and
gives the basic argument that it is uniquely qualified to be considered {the}
foundation of mainstream deductive mathematics. The ``coherent limit axiom''
characterizes the new theory among ZFC-like theories. This axiom plays a role
in recursion, but is implicitly assumed in mainstream work so does not provide
new leverage there. In principle it should settle set-theory questions such as
the continuum hypothesis.
