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". As noted elsewhere, some of the courses -- set theory, axiomatic geometry, statement logic, etc -- came to be encompassed in 1958 (!) under that rather silly label, "The New Mathematics". (Silly, because some math was "old", but had been forced out by educators.)

Here is a brief descriptive list of some of the "foundational schools":

• Logistic, from work of Gotthold Frege to Bertrand Russell and Alfred North Whiteheld; to derive all Mathematics from Logic.
• Formalism, championed by the German mathematician, David Hilbert (x-y), arguing that "mathematics is a formal game-like procedure of manipulating notation". In particular, Formalism allowed NONCONSTRUCTIVE PROOFS, in absence of CONSTRUCTIVE PROOFS.
• Intuitionism, championed by the Dutch mathematician, Luitzen Jan Egbert Brouwer (x-y), arguing that we have a natural intuition of the integers, from which all else proceeds. Intuitionism claims to recognize only CONSTRUCTIVE PROOFS.

I note three events in "Foundational" history.

1. Hilbert showed that all of Mathematics could be mapped into Arithmetic in manner of coordination in Analytic Geometry. So, if Arithmetic could be PROVEN CONSISTENT, so would all of Mathematics. But Kurl Gödel PROVED THAT ARITHMETIC CANNOT BE PROVEN COMPLETE AND CONSISTENT BY THE FINITARY METHODS DESIRED BY HILBERT -- dooming hopes of all above schools.
2. But Gergard Gentzen PROVED THAT CONSISTENCY AND COMPLETENESS OF ARITHMETIC CAN BE PROVEN BY A TRANSFINITARY METHOD -- which few are willing to accept.
3. xxxx PROVED THAT MUCH OF MATHEMATICS CAN BE FOUNDED CONSTRUCTIVELY -- but few feel happy about this.

Years after my 1957 Foundational Lectures, I've concluded that the SUFFICIENCY ASSUMPTIONS implicit in the above FOUNDATIONS are too much. But I believe that we can single out certain NECESSARY CONDITIONS FOR FOUNDING MATHEMATICS, such as CLOSURE and RECURABILITY.

Hence, my position of NECESSITARIANISM. Dig or Redig.