Thales of Miletus (c. 624-547 BC) is credited with creating "synthetic" geometry (as opposed to "analytic" geometry, credited to René Descartes (1596-1650)), and thereby "axiomatics". I call this "topdown" math, since it "descends" from "axioms" or "postulates" homologous to the Ten Commandments Moses brought down from Mount Ararat.

Pythagoras of Samos (c. 580-496), pupil of Thales, created "figurate geometry" (alternatively, "geometric numbers"), which is essentially "generative". I call this "bottomup", since it is "built up" from basic units, which Pythagoras called "gnomons".

In general, axiomatics and generatics are, currently, the two principal methods for teaching and learning mathematics.

There is a third notable foundation for mathematics -- the group-theoretic or transformation of conservation method, originating with German mathematician, Felix Klein (1849-1925). However, instead of introducing mathematics via this method, it is most useful in understanding mathematics once presented. For example, Klein showed us that the basic property of Euclidean geometry, namely, CONGRUENCE (matching of structures), is CONSERVED by "the Euclidean group", composed of 3 transformations, namely, translation (push), reflection (flip), rotation (turn). Klein could then defined a vast mathematical subject in a single, simple declarative sentence: Euclidean geometry is the study of all properties conserved under the transformations of the Euclidean Group. Nuff said. (Similarly, The Special Theory of Relativity can be simply defined as "the study of all physical properties conserved under the Lorentz-Einstein-Poincaré Group.)

Those needing elucidation, can understand "analogic" and "digital" by considering two prevailing forms of clocks. The traditional clock is "analogic" because of THE ANALOGY BETWEEN THE HANDS SWEEPING AROUND THE CLOCK-FACE-WITH-NUMBERS, on the one hand, AND THE PASSAGE OF TIME, on the other. By contrast, the more modern type of clock is "digital" because DIGITS "measuring" the time pop up in a slot on the clock face.

The point of this file is My Argument: GENERATICS IS BETTER THAN AXIOMATICS FOR TEACHING AND LEARNING MATHEMATICS. I'll list advantages:

1. we learn to speak LANGUAGE generatively, with the gnomonic phoneme, basic unit of sound;
2. we learn to read and write LANGUAGE generatively, with gnomic alphabetic letters, approximately phonetic units of signs;
3. teaching math generatively UNIFIES "the three R's: reading, writing, 'rithmetic" by using the same method (generatic) for all three, whereas axiomatic math bifurcates "the three R's";
4. the graphical ploy ("one picture is worth a thousand words") is more limited in synthetic geometry than in analytic geometry (at most, quasi-axiomatic), whereas the digital aspect of GENERATICS lends extensively to GRAPHIC exploitation;