Pythagoras of Samos (580-496 BC) was the pupil of Thales of Miletus (624-547 BC). Thales was the first to attempt GEOMETRIC PROOFS, initiating the AXIOMATIC PRESENTATION OF MATHEMATICS, a top-down procedure. In contrast, Pythagoras initiated the GENERATIC PRESENTATION OF MATHEMATICS, a bottoms-up procedure, and similar to ALPHABETIC writing, which the Greeks promoted.

In particular, Pythagoras developed "figurat geometry": he built geometric structures from dots or small stones. Yes, Pythagoras anticipated our model situation of digital-analogic structures. Lines and squares and planes and cubes and 3-D manifolds are CONTINUOUS, described in jargon as "analogic". But ARITHMETIC is "digital", or DISCONTINUOUS or DISCRETE, composed of "gnomons" or BASIC ELEMENTS, just as computer graphic is composed of tiny dots of PIXELS.

In speaking of numbers such as 4, 9, 16, 25, we use the label "squares", because Pythagoras taught us to build them as squares of dots. He taught us the word "cube", for the numbers 1, 8, 27, 64, 125, etc., from the 3-D structure that can be built from basic elements.

Pythagoras also spoke of the "triangular numbers", "pentagonal numbers", "hexagonal numbers", etc.

Pythagoras taught us the concept of the gnomon as the building-block of number patterns. (The word also refers to the device on the sundial which casts "the shadows of the hours". The word "gnomon" is also related to words for "knowing". Thus "gnostic" means "secret knowledge".)

Pythagoras or the Pythagoreans created the mathematical concepts of PRIME and COMPOSITE NUMBERS. A number is a PROPER FACTOR of another number if it is neither 1 nor the number itself. A PRIME NUMBER HAS NO PROPER FACTORS -- only 1 and the number itself. Examples: 2, 3, 5, 7, 11, 13, etc. A COMPOSITE NUMBER -- such as 6, 15, 21, 35, etc. -- has AT LEAST ONE PROPER FACTOR. Thus, The Pythagoreans inspired the search for ATOMS, MOLECULES, CHEMICAL ELEMENTS, BASIC MACHINES, GENES, CHROMOSOMES, in modern science.

This is so simple and self-evident that I've taught Preschool Children to emulate Pythagorean or Figurate Geometry by using bottle caps (or pebbles or pegs in a pegboard or tiles on the floor or wall). One BONUS is that children can DISCOVER THE COMMUTATIVE AND ASSOCIATIVE LAWS FOR ADDITION AND MULTIPLICATION -- usually just dictated to them -- by ROWS and TABLES of dots. I demonstrate with asterisks for dots or bottle caps or pebbles.

This demonstrates the ADDITIVE COMMUTATIVITY of 2 + 3 = 3 + 2 = 5:

*  * | *  *  *  -->  *  *  * | *  *  -->  *  *  *  *  * 

This demonstrates the MULTIPLICATIVE COMMUTATIVITY of 2 X 3 = 3 X 2 = 6:

*  *  -->  *  *  *  -->  *  *  *  *  *  *
*  *       *  *  *
*  *                                     

Pythagoras taught us that the odd number is the gnomon of the square. That is, squares are generated by summing odd numbers. Thus, 1 + 3 = 4 = 2 x 2, the "square of 2". 1 + 3 + 5 = 9 = 3 x 3, "square of 3". 1 + 3 + 5 + 7 = 16 = 4 x 4. 1 + 3 + 5 + 7 + 9 = 25 = 5 x 5. Etc.

*   |  *  *   |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |  *  *   |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |         |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |         |           | *  *  *  *  | *  *  *  *  *  |
    |         |           |             | *  *  *  *  *  |

The triangular numbers are 1, 2, 3, 4, 5, etc., -- the "counting numbers" -- because building rows of one bottle cap (or dot), a row of two, a row of three etc., builds larger and larger triangles.

    *    |     *       |     *     |     *     |     *     |     *     |
         |    * *      |    * *    |    * *    |    * *    |    * *    |
         |             |   * * *   |   * * *   |   * * *   |   * * *   |
         |             |           |  * * * *  |  * * * *  |  * * * *  |
         |             |           |           | * * * * * | * * * * * |
         |             |           |           |           |* * * * * *|

(Do you recognize the middle triangle? Look at it upside down. Thanks how, as a bowler, you see the ten-pin set on a bowling alley.)

The pentagonal numbers are 7, 15, 26, 40, 57, etc., because dots or caps of these build up larger and larger pentagons.

But the discovery that THE DIAGONAL OF A SQUARE CANNOT BE REPRESENTED BY A NUMBER OR RATIO OF NUMBERS created a crisis for the Pythagoreans and an opportunity for their enemies.

The story was told that one Pythagorean, Hippias (x-y), discovered this "gate" problem and spread the information. So, it was alleged that his follow Pythagoreans drowned him for this "heresy". But scholars believe this story was made up and spread by Pythagorean enemies, since the Pythagorean forbade the taking of life -- even of suicide. (The present Hippocratic Oath of physicians derived from an OATH taken by Pythagorean physicians, who ofen used music to calm the psychotic or neurotic.)

But, as we see in another file, a philosophical enemy, ZENO (x-y), created "PARADOXES" attack the Pythagorean Figurate Geometry. These paradoxes were only resolved in modern times.