The literature shows three FOUNDATIONS FOR MATHEMATICS:

1. AXIOMATIC -- deriving from Thales of Miletus (c.624-547BC), exemplified in Euclid's Elements of Geometry.
2. GENERATIVE -- deriving from the pupil of Thales, Pythagoras (c.589-496BC).
3. GROUP-THEORETIC -- deriving from German mathematician, Felix Klein (1849-1925).

These FOUNDATIONS are especially useful in THE TACTICS OF MATHEMATICS. I use the generative method to show how a comprehensive CURRICULUM can be developed which fulfils what the great Henri Poincaré (1854-1917) may have meant by saying, "Mathematics has been arithmetized." (http://andrew.cmu.edu/~cebrown/notes/leivant.html )

And I show how the generative draws upon THE WAY WE LEARN LANGUAGE AND READING, hence, UNIFIES "THE THREE R'S" (where the AXIOMATIC bifurcates them), and may give an advantage to girls and women, since they are often superior to boys and men in language.

But I'm dissatisfied with these FOUNDATIONS, because they fail

1. to charlotte with my educational ideas -- as in "African Violet" Education;

2. to deal with "Math as Patterns-in-Patterns-in-..." and the human need to feel that THEIR LIVES FIT A PATTERN;

3. to deal with my observation that (in every endeavor!) we are "TACTICS RICH & STRATEGY POOR";
4. to deal with the (FORMALIC) role of Mathematics in the semiotic hierarchy (in the "theory of signs");
5. especially, to deal with the charlotte of Canadian mathematician, Z. A. Melzak, that HOMINID BECAME HUMAN BY INTERNALIZING THE BYPASS STRATEGY (turning a difficult or impossible problem into a problem hominid or human knows how to solve).

I, therefore, DECLARE:

MATHEMATICS IS FORMALICALY PATTERNED STRATEGY!

Briefly, MATHEMATICS

1. begins with HUMAN STRATEGIES;

2. USES FORMALIC CONSTRAINTS TO YIELD PATTERNS;

3. HENCE, TRANSFORM FORMALIZED PATTERNS INTO WIN-WIN TOOLS

4. which are TESTABLE BY THE FIGURE&GROUND STRATEGY.

In jargon:

MATH = BYPASS + STRATEGY + FORMALICS + PATTERNS + WINWIN + FIGURE&GROUND,

so that

MATH Þ LIFE.

This charlottes with another finding. The eminent mathematical-logician, Haskell Curry (in his Foundations of Mathematical Logic), says (p. 8), "There are two main types of opinions in regard to the nature of mathematics. We shall call these contensivism and formalism." Formalism (Hilbert is the main spokesman for this viewpoint) regards MATH AS A "GAME" FORMALIZED OR TAKING SHAPE BECAUSE OF RULES. Take away the rules of Checkers or Chess and what is left? Nothing two people can agree on; nothing so that two people will feel they are "talking about the same thing". Similarly, with THE FORMALIC CONCEPTION OF MATHEMATICS.

But, as I understand Curry, CONTENSIVISM SAYS RULES HAVE BEEN APPLIED TO "SOMETHING". TAKE AWAY THE RULES, and THERE'S STILL "SOMETHING" PEOPLE CAN AGREE ON.

So, in this sense, I'm a CONTENSIVIST [yes, Hamlet, apparently from "form with content"]: RULES APPLIED TO PATTERNS. TAKE AWAY THE RULES, AND PEOPLE CAN STILL AGREE ON "SEEING" THE SAME RESIDUE PATTERN!

Hey! That means that PROTO-MATH WAS ALWAYS THERE -- AS LONG AS HUMANS OR HOMINIDS SENSED PATTERNS, EVEN AS BASIC AS PERCEPTIONS AND "THINKS". SO, WE'RE ALL PROTO-EDUCATED IN MATHEMATICS. AND THIS IS HOW WE CAN HELP CHILDREN!

Charlotte?

DIG OR REDIG!