thinking up new research problems
different from solving
Missionaries and Cannibals problem.

explain theory formation: analyze discoveries
primes, given factoring/divisors.

"how in the world could you discover prime numbers"→ "how...divisors-of"
	accomplished by citing heur: if f is int, investigate f↑-↑1(int set).

Begin to get a handle on what it might mean to analyze a dsicvovery: keep reducing it.

Get divisors-of by: inverse of mult.

Get mult by: completing the square

Reverse the direc of these reducs: get a plausible scenario for discov.
Gather a large body of heurs., use them to generate new concepts, make discovs.

In the synthesis direc, the tree is very broad, the "space" is very big.
How to know which way to go (e.g.: from Divisors, go to primes or Max-divis)
	Local judgment criteria.
No specific goal: overall quality of the behavior.

Max-divis: converse of Primes; hitherto unstudied explicitly.
The form of these numbers is necessarily 
2⊗Aa1⊗*  3⊗Aa2⊗* 5⊗Aa3⊗*...   p⊗Bk⊗*⊗Aak⊗*,  where the
p⊗Bi⊗*'s are the first k consecutive primes, 
so we must start with 2 and may not skip any prime,
and the exponents a⊗Bi⊗*
decrease   with  i,  and   the  ratio  of   (a⊗Bi⊗*+1)/(a⊗Bj⊗*+1)  is
approximately   (as   closely   as    is   possibe   for    integers)
log(p⊗Bj⊗*)/log(p⊗Bi⊗*). For  example, a typical  divisor-rich number
is
n=2⊗A8⊗*3⊗A5⊗*5⊗A3⊗*7⊗A2⊗*11⊗A2⊗*13⊗A1⊗*17⊗A1⊗*19⊗A1⊗*23⊗A1⊗*29⊗A1⊗*31⊗A1⊗*37⊗A1⊗*41⊗A1⊗*43⊗A1⊗*47⊗A1⊗*53⊗A1⊗*.
The progression of its exponents+1 (9 6 4 3 3 2 2 2 2 2 2 2 2 2 2 2)
is  about as  close  as one  can  get  to satisfying the  "logarithm"
constraint.   
By that I mean that 9/6 is close to log(3)/log(2); that 2/2 is close to
log(47)/log(43), etc.  Changing any exponent by plus or minus 1 would make
those ratios worse than they are.
This  number n <point to n value> has about 4 million divisors.
The "AM  Conjecture" is that no  number smaller than n  has
that many divisors.
This is so far the only piece of interesting mathematics, previously unknown, 
that was directly motivated by AM. 

Program it: decide concepts and heurs.

For AM:
  Concepts: Static (Sets, Conjec), Dynamic(Union, Compose).
	Facets: initial (defn, algs, dr, intu, ¬int), potential (¬exs, ties)
  Heur: Guide (If few exs, genlize), Fillin (consider exs of related concepts),
		Judge (Composn. is int if D=R; if props of f1/f2 preserved).
	CONSTRAINT: demonic, only considered when plausible. Soln: attach to concepts.
	Advantage: questionaire-like structuring to elicit heurs. in vacuoo.
	CONSTRAINT: Must be abl to work together (eg. Compos(x,y): all 3 sets of heurs)
		Solution: cooperating knowl. sources: pred calc/conjoin; prods/combine.
  Control: expand knowl = grow concepts and relnships = pick a facet of a concept and
	fill it in. Use heurs to help pick and help fill in and decide when to split.
  Filling in some some parts (Algs, Defn) means automatic pgmming: BEINGs.

NOT specific to MATH, but to thy formation, research, in any science.
	Just choose new concepts, facets, heurs.

Why MATH: no uncertainties in data, domain expert = pgmmer = me,  no single goal.
	(not limited to explaining fixed, given experimental data; always OK to give up).

Math is an EMPIRICAL science.
	Opposite to texts. Observe, notice, describe formally, finalize, predict.
	Awkward stateent of relnship: then try a new defn.

Belief: changing knowledge base, DON'T update. Handle conflicts at paradox-time.

Gap: English convincing VS LISP convinving. 
	We can talk about more than we really understand.

Next step: apply this to other domains of mathematics, extend its abilities
	Make it a more-rounded mahtematician. 
	Do experiments on it: remove, add, modify     concepts and heurs.
	2 goals: good performance, really at the level of research math;
			Determine the heurs needed, how used, etc.;
				isolate those which are genl. to all thy formation.