RECURSION is the most POWERFUL TOOL in MATHEMATICS. You dig RECURSION, don't you?

Sure you do. Your savings account accumulates compound interest recursively. MONEY PRESENT accumulates INTEREST, and its ADDITION INCREASES MONEY PRESENT, feeding back to earn new INTEREST. In general, RECURSION IS INPUT TRANSFORMED INTO OUTPUT, BECOMING NEW INPUT. ("Gee! Like sssssexxxxxxxxxx!")

RECURSION CREATES OUR ARITHMETIC OPERATIONS, AND MODELS MANY OR MOST OF OUR PHYSICAL PROCESSES. (Niels K. Jerne, 1984 Nobelist in Physiology and Medicine, used Noam Chomsky's recursive model of language, "generative grammar", to explain the human immune system, equating "components of a generative grammar ... with various features of protein structures".)

But RECURSION was begot in Eden -- ferinstance, in those sentences of Eve's which Adam tried to parse.

Eve says, "Rocky Racoon, who chased Ollie Otter back into the ocean where his friend Sammy Seal played, washed his breakfast in the pool of fresh water." Eve sat back smiling, waiting for Adam to parse that one.

Eve was a talky girl -- the first of generations of talky girls. Who have talked our male ears off. And, threading a needle, sewed them back on. Here's one talky little girl reviewing The Wizard of Oz: "Dorothy, who met the Wicked Witch of the West in Munchkin Land where her wicked witch sister was killed, liquidated her with a pail of water."

Dig? Whenever you put sentences inside sentences you are RECURSING. MIT mathematical linguist Noam Chomsky hypothesized that humans generate language recursively, rebutting behavioral psychologist B. S. Skinner's claim that humans learn language by trial-and-error associations. Chomsky showed that associationism is modeled probabilistically by multiple-order Markhov chains. A chosen word determines the probabilities of the words that follow. Mathematical psychologist George Miller found that (given a chosen word), "on the average", four options exist for the grammatical category of the next word. In a first-order Markhov chain, a child must learn 4 x 4 = 4² = 16 associations (four contexts times four next words); for a second-order chain (determining "odds" on the next two words), 4 x 4 x 4 = 4³ = 64 associations; etc. As context increases, the number of learned associations QUADRUPLES.

Consider again that sentence of the talky little girl): "Dorothy, who met the Wicked Witch of the West in Munchkin Land where her wicked witch sister was killed, liquidated her with a pail of water." Subject and predicate of "Dorothy liquidated her with a pail of water" are separated by an eighteen-word clause -- invoking an eighteenth-order Markhov chain of 4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4 = 4¹⁸ associationss. Learning to speak such a sentence by Skinner's trial-and-error model requires nearly 69 billion associations. The child should live so long!

Repeating, RECUSION was begot, among other origins, in Eve's convoluted sentences

And when Eve and Adam were exiled from the Garden of Eve and discovered how to BEGAT children, RECURSIVENESS thrived, from Prehistory to History, in the BEGATING OF THEIR DESCENDANTS.

As recorded in the Fifth Chapter of Genesis in The Bible:

1. Adam begat Seth;
2. Seth begat Enos;
3. Enos begat Cainan
4. Cainan begat Mahaleel;
5. Mahaleel begat Jared;
6. Jared begat Enoch;
7. Enoch begat Methusaleh.

(And, as Sportin' Life sings, in the Gershwins' Porgy and Bess, "Methusaleh lived nine hundred years. But who calls that livin when no gal will give in to no man who lived nine hundred years?")

Let's designate B(_) as the begat-operator. Thus:

1. B(Adam) = Seth;
2. B(Seth) = Enos, that is, B(B(Adam)) = Seth;
3. B(Enos)B(B(Seth)) = Cainan, that is, B(B(B(Adam))) = Cainan;
4. B(Cainan) = Maheel, that is,B(B(B(B(Adam)))) = Maheel;
5. B(Maheel) = Jared, that is, B(B(B(B(B(Adam))))) = Jared;
6. B(Jared) = Enoch, that, B(B(B(B(B(B(Adam)))))) = Enoch;
7. B(Enoch) = Methusaleh, that is, B(B(B(B(B(B(B(Adam))))))) = Methusaleh.

Then, B(B(B(B(B(B(B(Adam))))))) = Methusaleh, the 7th generation begot from Adam.

(E. Y. Harburg and Burton Lane wrote a "Begin the Begat" song for Finian's Rainbow, announcing, "They begat the misbegotten GOP!")

Do you recognize the MATHEMATICS in this? You should! Just by CHANGING THE LABEL FROM "B" (FOR "BEGAT") TO "S" (FOR "SUCCESSOR"), you find you are working with THE SUCCESSOR OPERATOR WHICH GENERATES ("BEGOTS") ALL THE NATURAL (COUNTING) NUMBERS OF ARITHMETIC -- THE BASIS ("BEGOTTER") OF ALL ARITHMETIC -- AND (as David Hilbert (1862-1943) taught us) YOU CAM MAP ALL MATHEMATICS INTO ARITHMETIC ("BEGOTTER OF ALL MATH"). Shazamm! In the "BEGATS", you have the "SEED" of ALL MATHEMATICS!

Wotsomatter? Were you sleeping when Teecher did this in Class? Or didn't Teecher do it? Most don't; some from stubborness, most from ignorance. (I strut my stuff ONLINE to Teech Teechers.)

Looky! The SUCCESSOR OPERATOR is S(n) = n + 1. Wazzatmeen? "The Successor of any number n is n and one more!"

Before expanding this, I must point out (or remind you) that WE KNOW THE NATURAL NUMBERS BY THEIR NICKNAMES (so's we don't have to spiel out a long moniker, as if they're monarchs). The nickname of 0 + 1 is simply 1; the nickname of (0 + 1) + 1 = 0 + 1 + 1 is simply 2. The nickname of ((0 + 1) + 1) + 1 = 0 + 1 + 1 + 1 is simply 3; etsettery. Dig?

So we start GENERATING ("BEGATTING") THE NATURAL NUMBERS, STARTING WITH ZERO = 0. (For 0 is "The Adam" of THE NATURAL NUMBER GENERATIONS.)

1. S(0) = 0 + 1 = 1 ("nickname");
2. S(1) = 1 + 1 = 2 ("nickname"), that is, S(1) = S(S(0)) = 2;
3. S(2) = 2 + 1 = 3, that is, S(2) = S(S(1)) = S(S(S(0))) = 3;
4. S(3) = 3 + 1 = 4, that is, S(3) =S(S(2)) =S(S(S(1))) = S(S(S(S(0)))) = 4;
5. S(4) = 4 + 1 = 5, that is, S(S(S(S(S(0))))) = 5;
6. S(5) = 5 + 1 = 6, that is, S(S(S(S(S(S(0)))))) = 6;
7. S(6) = 6 + 1 = 7, that is, S(S(S(S(S(S(S(0))))))) = 7.

Thus, 7 is the seventh generation of 0 just as Methusaleh is the seventh generation of Adam. Dig?

GENERATING THE NATURAL NUMBERS is what I call "RECURSION IN THE FIRST DEGREE". SECOND DEGREE: GENERATING ADDITION (+) OF TWO NATURAL NUMBERS (a, b) BY RECURSION ON THE SUCCESSOR OPERATION: a + S(b) Þ S(a + b). This GENERATES ANY SUM, whatever. Thus, S(4 + 3) = 4 + S(3).

• If you know that 4 + 3 = 7, then (from understanding THE SUCCESSOR OPERATOR) S(4 + 3) = S(7) = 8 = 4 + S(3) = 4 + 4.
• If you don't readily know this, you always can -- in desperation -- start GENERATING that left term above (S(4 + 3) in a CHAIN OF SUCCESSIONS:
  1. S(4 + 0) = 4 + S(0) = 4 + 1 = 5 -- knowing the nickname of 4 + 1.
  2. Then, S(4 + 1) = 4 + S(1) = 4 + 2 = 4 + (1 + 1) = 6.
  3. Then, S(4 + 2) = 4 + S(2) = 4 + 3 = 4 + ((1 + 1) + 1) = 6.
  4. Finally, S(4 + 3) = 4 + S(3) = 4 + (((1 + 1) + 1) + 1) = 8

GENERATING MULTIPLICATION FROM ADDITION IS "RECURSION IN THE THIRD DEGREE": a · 1 = a; S(a · (b + 1)) = a + S(a · b). This GENERATES ANY PRODUCT, whatever.

AS "RECURSION IN THE FOURTH DEGREE", THE OTHER PRIMARY ARITHMETICAL OPERATION CAN BE DEROVED BY APPLING RECURSION TO MULTIPLICATION.

Yes, Ophelia, "it's even bigger than this". There's a very advanced field of Mathematics which developed in the second half of the 20th century, recursive set theory (a.k.a. recursive function theory).

A SET (say, "THE SQUARES", of natural numbers or integers) is RECURSIVELY ENUMERABLE IF, AND ONLY IF, WE HAVE AN ALGORITHM FOR GENERATING EVERY MEMBER OF THE SET. (For "the squares", we have an obvious algorithm: TAKE THE SET OF NATURALS OR INTEGERS; MULTIPLY ANY MEMBER BY ITSELF; IT IS A SQUARE, AND NO SQUARE ESCAPES FROM THIS.)

A SET IS RECURSIVE IF IT IS RECURSIVELY ENUMERABLE AND WE HAVE AN ALGORITHM FOR GENERATING THIS SET'S COMPLEMENT: ELEMENTS NOT IN THE SET. THERE ARE RECURSIVELY ENUMERABLE SETS WHICH ARE NOT RECURSIVE.

This, and other goodies of the Theory, provide profound tools for articulating and extending Mathematics. ALL FROM RECURSION, which STARTED IN EDEN (in Eve's convoluted sentences) AND EAST OF EDEN (in begating of children). And RECURSION BEGAT ARITHMETIC WHICH MAPS INTO ALL OF MATHEMATICS.

So, "MATH WAS BEGOT IN EDEN WHEN ADAM NAMED THE ANIMALS, EVE NAMED THE FLOWERS." ALLTHEN.