The term "foundation"is thus used to explain a given subject, in this case: MATHEMATICS. That is, what constitutes mathematics.

In SUMMER 1957, I helped organized and hold the first National Science Foundation Institute for High School Teachers of Mathematics to be held in Puerto Rico -- at The American University of Puerto Rico, San Germán, P. R. I gave a series of lectures "The Foundations of Mathematics".

I became sceptical about all the standard candidates for "Foundations". Typically, they are explicitly or implicitly AXIOMATICA. The foundational tenets offered are NECESSARY and SUFFICIENT.

So I'm offering some SUFFICITARIAN FOUNDATIONS. They seem to be present in all the MATH I've thought about:

• RECURSIVE, in the sense as used in RECURSIVE SET THEORY:
  1. The set constituting a given type of MATH is generated by a rule that captures all of it -- labeled "a RECURSIVELY ENUMERABLE SET".
  2. The rule also RULES OUT what we don't want in the set, that is, it CONSTRUCTS THE COMPLEMENTARY SET OF WHAT THE FIRST SET IS NOT.
  3. These two conditions PROVIDE FOR A RECURSIVE SET.
     CLOSURE ("All in the Family") for all involved sets. (The above provides for closure. But not everything is recursive).
     Is this enough? What do you say?