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␈↓ ↓H␈↓␈↓ ∧∨␈↓∧␈↓&Automated Math Theory Formation␈↓)αβ␈↓

␈↓ ↓H␈↓␈↓ ¬;␈↓∧Thesis Description␈↓

␈↓ ↓H␈↓␈↓ ¬m␈↓↓Douglas B. Lenat␈↓
␈↓ ↓H␈↓␈↓ ¬;Arti≡cial Intelligence Lab
␈↓ ↓H␈↓␈↓ ¬←Stanford University
␈↓ ↓H␈↓␈↓ ¬2Stanford, California 94305


␈↓ ↓H␈↓Scientists␈α∂often␈α∂face␈α∂the␈α∂di≠cult␈α∂task␈α⊂of␈α∂formulating␈α∂research␈α∂problems␈α∂which␈α∂must␈α∂be␈α⊂soluble␈α∂yet
␈↓ ↓H␈↓nontrivial.␈α In␈αany␈αgiven␈αbranch␈αof␈αscience,␈αit␈αis␈αusually␈αeasier␈αto␈αtackle␈αa␈αspeci≡c␈αgiven␈αproblem␈αthan
␈↓ ↓H␈↓to␈αpropose␈α
interesting␈αyet␈αmanagable␈α
new␈αquestions␈α
to␈αinvestigate.␈α For␈α
example,␈αcontrast␈α
␈↓βsolving␈↓␈αthe
␈↓ ↓H␈↓Missionaries and Cannibals problem with the more ill-de≡ned reasoning which led to ␈↓βinventing␈↓ it.

␈↓ ↓H␈↓For␈α∂my␈α⊂dissertation,␈α∂I␈α∂am␈α⊂investigating␈α∂creative␈α⊂theory␈α∂formation␈α∂in␈α⊂mathematics:␈α∂how␈α⊂to␈α∂propose
␈↓ ↓H␈↓interesting␈αnew␈αconcepts␈αand␈αplausible␈αhypotheses␈αconnecting␈αthem.␈α The␈αexperimental␈αvehicle␈αof␈αmy
␈↓ ↓H␈↓research␈αis␈α
a␈αcomputer␈αprogram␈α
called␈α␈↓↓AM␈↓␈α(for␈α
␈↓↓␈↓&A␈↓)αβ␈↓utomated␈α␈↓↓␈↓&M␈↓)αβ␈↓athematician),␈αwhich␈α
carries␈αout␈αsome␈α
of
␈↓ ↓H␈↓the␈α∩activities␈α∩involved␈α∩in␈α∩mathematical␈α∩research:␈α∩noticing␈α∩simple␈α∩relationships␈α∩in␈α∩empirical␈α⊃data,
␈↓ ↓H␈↓formulating␈α⊃new␈α∩de≡nitions␈α⊃out␈α∩of␈α⊃existing␈α⊃ones,␈α∩proposing␈α⊃some␈α∩plausible␈α⊃conjectures␈α∩(and,␈α⊃less
␈↓ ↓H␈↓importantly, sometimes proving them), and evaluating the aesthetic "interestingness" of new concepts.

␈↓ ↓H␈↓Before␈α∩discussing␈α∩how␈α∩to␈α∩␈↓βsynthesize␈↓␈α∩a␈α∩new␈α∩theory,␈α∩consider␈α∩brie∨y␈α∩how␈α∩to␈α∩␈↓βanalyze␈↓␈α∩one,␈α∩how␈α∩to
␈↓ ↓H␈↓construct␈α
a␈α
plausible␈α
chain␈α
of␈αreasoning␈α
which␈α
terminates␈α
in␈α
a␈αgiven␈α
discovery.␈α
One␈α
can␈α
do␈α
this␈αby
␈↓ ↓H␈↓working␈α
backwards,␈α
by␈α
reducing␈α
the␈α
creative␈α
act␈αto␈α
simpler␈α
and␈α
simpler␈α
creative␈α
acts.␈α
 For␈αexample,
␈↓ ↓H␈↓consider␈α
the␈α
concept␈α
of␈α
prime␈α
numbers.␈α
 How␈α
might␈α∞one␈α
be␈α
led␈α
to␈α
de≡ne␈α
such␈α
a␈α
notion?␈α∞Notice␈α
the
␈↓ ↓H␈↓following plausible strategy:

␈↓ ↓H␈↓␈↓ αX"If␈αf␈αis␈αa␈αfunction␈αwhich␈αtransforms␈αelements␈αof␈αA␈αinto␈αelements␈αof␈αB,␈αand␈αB␈αis
␈↓ ↓H␈↓␈↓ αXordered,␈α∂then␈α⊂consider␈α∂just␈α⊂those␈α∂members␈α⊂of␈α∂A␈α⊂which␈α∂are␈α⊂transformed␈α∂into
␈↓ ↓H␈↓␈↓ αX␈↓βextremal␈↓ elements of B.  This set is an interesting subset of A."

␈↓ ↓H␈↓When␈αf(x)␈αmeans␈α"factors␈αof␈αx",␈αand␈αthe␈αordering␈αis␈α"by␈αlength",␈αthis␈αheuristic␈αsays␈αto␈α
consider␈αthose
␈↓ ↓H␈↓numbers␈αwhich␈αhave␈αa␈αminimal␈↓	1␈↓␈αnumber␈αof␈αfactors␈α
--␈αthat␈αis,␈αthe␈αprimes.␈αSo␈αthis␈αrule␈αactually␈α
␈↓βreduces␈↓
␈↓ ↓H␈↓our␈α∪task␈α∪from␈α∪"proposing␈α∪the␈α∪concept␈α∪of␈α∪prime␈α∪numbers"␈α∪to␈α∪the␈α∪more␈α∪elementary␈α∪problems␈α∩of
␈↓ ↓H␈↓"inventing factorization" and "discovering cardinality".

␈↓ ↓H␈↓But␈α
suppose␈αwe␈α
know␈αthis␈α
general␈αrule:␈α
"If␈αf␈α
is␈αan␈α
interesting␈αfunction,␈α
consider␈αits␈α
inverse."␈αIt␈α
reduces
␈↓ ↓H␈↓the␈α
task␈αof␈α
discovering␈αfactorization␈α
to␈αthe␈α
simpler␈αtask␈α
of␈αdiscovering␈α
multiplication.␈α Eventually,␈α
this
␈↓ ↓H␈↓task␈α⊂reduces␈α⊂to␈α⊂the␈α⊂discovery␈α⊂of␈α⊂very␈α⊂basic␈α⊂notions,␈α⊂like␈α⊂substitution,␈α⊂set-union,␈α⊂and␈α⊂equality.␈α∂ To
␈↓ ↓H␈↓explain␈α∞how␈α∞a␈α
given␈α∞researcher␈α∞might␈α∞have␈α
made␈α∞a␈α∞given␈α∞discovery,␈α
such␈α∞an␈α∞analysis␈α∞is␈α
continued
␈↓ ↓H␈↓until that inductive task is reduced to "discovering" notions which the researcher already knew.

␈↓ ↓H␈↓Suppose␈α∂a␈α∂large␈α∂collection␈α∂of␈α∂these␈α∂heuristics␈α∞has␈α∂been␈α∂assembled␈α∂(e.g.,␈α∂by␈α∂analyzing␈α∂a␈α∂great␈α∞many
␈↓ ↓H␈↓discoveries,␈α⊂and␈α⊂writing␈α⊂down␈α⊂new␈α⊂heuristic␈α⊂rules␈α⊂whenever␈α⊂necessary).␈α⊂ Instead␈α⊂of␈α⊂using␈α⊂them␈α∂to
␈↓ ↓H␈↓␈↓βexplain␈↓␈α∩how␈α∩a␈α∩given␈α∩idea␈α∩might␈α∩have␈α∩evolved,␈α∩one␈α∩can␈α∩imagine␈α∩starting␈α∩from␈α∩a␈α∩basic␈α∩core␈α∩of
␈↓ ↓H␈↓knowledge and "running" the heuristics to ␈↓βgenerate␈↓ new concepts.

␈↓ ↓H␈↓Such␈α∀syntheses␈α∀are␈α∀precisely␈α∀what␈α∀AM␈α∀does.␈α∀ The␈α∀program␈α∀consists␈α∀of␈α∀a␈α∀corpus␈α∃of␈α∀primitive
␈↓ ↓H␈↓mathematical␈α∂concepts␈α∂and␈α∂a␈α∂collection␈α∂of␈α∞guiding␈α∂heuristics␈α∂(currently␈α∂a␈α∂couple␈α∂hundred␈α∂of␈α∞each).
␈↓ ↓H␈↓AM's␈α∩activities␈α∩all␈α⊃serve␈α∩to␈α∩expand␈α∩AM␈α⊃itself,␈↓	2␈↓␈α∩to␈α∩enlarge␈α∩upon␈α⊃a␈α∩given␈α∩body␈α∩of␈α⊃mathematical
␈↓ ↓H␈↓knowledge.␈α∪ To␈α∪cope␈α∀with␈α∪the␈α∪enormity␈α∀of␈α∪the␈α∪potential␈α∪"search␈α∀space"␈α∪involved,␈α∪AM␈α∀uses␈α∪its
␈↓ ↓H␈↓heuristics␈α∞as␈α∞judgmental␈α∞criteria␈α∞to␈α∞guide␈α∞development␈α∞in␈α∞the␈α∞most␈α∞promising␈α∞direction.␈α∂ It␈α∞appears
␈↓ ↓H␈↓that␈αthe␈α
process␈αof␈α
inventing␈αworthwhile␈α
new␈↓	3␈↓␈αconcepts␈αcan␈α
be␈αguided␈α
successfully␈αusing␈α
a␈αcollection
␈↓ ↓H␈↓of a few hundred such heuristics.

␈↓ ↓H␈↓Each␈α
concept␈αis␈α
represented␈α
as␈αa␈α
␈↓¬BEING␈↓␈↓	4␈↓,␈α
a␈αframe-like␈α
data␈αstructure␈α
with␈α
30␈αdi≥erent␈α
facets␈α
or␈αslots.
␈↓ ↓H␈↓The␈α∩types␈α∩of␈α⊃slots␈α∩include:␈α∩␈↓¬Examples,␈α⊃Definitions,␈α∩Generalizations,␈α∩Utility,␈α∩Analogies,␈α⊃Interestingness,
␈↓ ↓H␈↓¬Uninterestingness,␈↓␈α∩and␈α∩a␈α∩couple␈α∩dozen␈α∩others.␈α∩ The␈α∩␈↓¬BEINGs␈↓␈α∩representation␈α∩provides␈α∩a␈α∩convenient
␈↓ ↓H␈↓scheme␈αfor␈αorganizing␈αthe␈αheuristics;␈αfor␈αexample,␈αthe␈αfollowing␈αstrategy␈α≡ts␈αinto␈αthe␈α␈↓βExamples␈↓␈αslot␈αof
␈↓ ↓H␈↓the␈α␈↓βPredicate␈↓␈α
concept:␈α"If,␈αempirically,␈α
10␈αtimes␈α
as␈αmany␈αelements␈α
␈↓βfail␈↓␈αsome␈αpredicate␈α
P,␈αas␈α
␈↓βsatisfy␈↓␈αit,
␈↓ ↓H␈↓then␈αsome␈α␈↓βgeneralization␈↓␈α(weakened␈αversion)␈αof␈αP␈αmight␈αbe␈αmore␈αinteresting␈αthan␈αP".␈α AM␈αconsiders
␈↓ ↓H␈↓this suggestion after trying to ≡ll in examples of any predicate.␈↓	5␈↓

␈↓ ↓H␈↓AM␈α
is␈α
initially␈α
given␈α
a␈α
large␈α
collection␈α
of␈α
core␈αconcepts,␈α
with␈α
only␈α
a␈α
few␈α
slots␈α
≡lled␈α
in␈α
for␈α
each.␈α Its
␈↓ ↓H␈↓sole␈αactivity␈αis␈αto␈αchoose␈αsome␈αfacet␈αof␈αsome␈αconcept,␈αand␈α≡ll␈αin␈αthat␈αparticular␈αslot.␈α In␈αso␈αdoing,␈αnew
␈↓ ↓H␈↓notions␈αwill␈αoften␈αemerge.␈α Uninteresting␈αones␈α
are␈αforgotten,␈αmildly␈αinteresting␈αones␈αare␈αkept␈α
as␈αparts
␈↓ ↓H␈↓of␈αone␈αslot␈αof␈αone␈αconcept,␈αand␈αvery␈αinteresting␈αones␈αare␈αgranted␈αfull␈αconcept␈αstatus.␈αSuch␈αnew␈α␈↓¬Beings␈↓
␈↓ ↓H␈↓have␈α
dozens␈α
of␈α
blank␈α∞parts,␈α
hence␈α
the␈α
space␈α∞of␈α
possible␈α
actions␈α
(slots␈α∞to␈α
≡ll␈α
in)␈α
grows␈α∞rapidly.␈α
 The
␈↓ ↓H␈↓same␈αheuristics␈αare␈αused␈αboth␈αto␈αsuggest␈αnew␈αdirections␈αfor␈αinvestigation,␈αand␈αto␈αlimit␈αattention:␈αboth
␈↓ ↓H␈↓to grow and to prune.

␈↓ ↓H␈↓The␈αparticular␈α
mathematical␈αdomains␈α
in␈αwhich␈αAM␈α
operates␈αdepend␈α
on␈αthe␈α
choice␈αof␈αinitial␈α
concepts.
␈↓ ↓H␈↓Currently,␈α⊃AM␈α⊃begins␈α⊃with␈α⊃scanty␈α⊃knowledge␈α⊂of␈α⊃a␈α⊃couple␈α⊃hundred␈α⊃concepts␈α⊃which␈α⊃Piaget␈α⊂might
␈↓ ↓H␈↓describe␈α∞as␈α∞␈↓βprenumerical␈↓:␈α∞Sets,␈α∞substitution,␈α∞relations,␈α∞equality,␈α
and␈α∞so␈α∞on.␈α∞ In␈α∞particular,␈α∞AM␈α∞is␈α
not
␈↓ ↓H␈↓told␈α∃anything␈α∃about␈α∃proof,␈α∃single-valued␈α∃functions,␈α∀or␈α∃numbers.␈α∃With␈α∃this␈α∃basis,␈α∃AM␈α∀quickly
␈↓ ↓H␈↓discovered␈α↔elementary␈α↔numerical␈α↔concepts␈α↔(corresponding␈α↔to␈α↔those␈α↔we␈α↔refer␈α↔to␈α↔as␈α⊗cardinality,
␈↓ ↓H␈↓multiplication,␈α∩factors,␈α∩and␈α∩primes)␈α∩and␈α∩wandered␈α∩around␈α∩in␈α∩the␈α∩domain␈α∩of␈α∩elementary␈α⊃number
␈↓ ↓H␈↓theory.␈α⊂ Although␈α⊃it␈α⊂was␈α⊃never␈α⊂able␈α⊃to␈α⊂␈↓βprove␈↓␈α⊂the␈α⊃unique␈α⊂factorization␈α⊃theorem,␈α⊂AM␈α⊃actually␈α⊂did
␈↓ ↓H␈↓␈↓βconjecture␈↓␈αit.␈↓	6␈↓␈α"Discovering"␈αa␈αconcept␈αmeans␈αthat␈α(1)␈αAM␈αrecognized␈αit␈αas␈αa␈αdistinguished␈αentity␈α(e.g.,
␈↓ ↓H␈↓by␈α
formulating␈α
its␈α
de≡nition)␈α
and␈αalso␈α
(2)␈α
AM␈α
decided␈α
it␈αwas␈α
worth␈α
investigating␈α
(either␈α
because␈αof
␈↓ ↓H␈↓the interesting way it was formed, or because of surprising preliminary empirical results).

␈↓ ↓H␈↓AM␈αwas␈αnot␈αable␈αto␈αdiscover␈αany␈α"new-to-Mankind"␈αmathematics␈αpurely␈αon␈αits␈αown,␈αbut␈α␈↓βhas␈↓␈αdone␈αso
␈↓ ↓H␈↓when␈α↔working␈α↔as␈α↔a␈α_co-researcher␈α↔with␈α↔a␈α↔human.␈↓	1␈↓␈α_AM␈α↔noticed␈α↔simple␈α↔new␈α_concepts␈α↔which
␈↓ ↓H␈↓mathematicians␈α⊂had␈α⊂overlooked,␈α⊂but␈α⊂AM␈α⊂by␈α⊂itself␈α∂was␈α⊂not␈α⊂able␈α⊂to␈α⊂precisely␈α⊂formulate␈α⊂and␈α∂prove
␈↓ ↓H␈↓interesting␈α
statements␈αabout␈α
those␈α
concepts.␈α I␈α
conclude␈α
that␈αa␈α
synergetic␈α
AM--human␈αcombination␈α
can
␈↓ ↓H␈↓sometimes produce better research than either could alone.

␈↓ ↓H␈↓The␈α
main␈α
di≠culty␈α
with␈α
AM␈α
at␈α
present␈α
is␈α
getting␈α
it␈α
to␈α
accurately␈α
judge␈α
␈↓βa␈α
priori␈↓␈α
the␈α
value␈α
of␈αeach␈α
new
␈↓ ↓H␈↓concept,␈α∂to␈α∂quickly␈α∞lose␈α∂interest␈α∂in␈α∂concepts␈α∞which␈α∂aren't␈α∂going␈α∂to␈α∞develop␈α∂into␈α∂anything.␈α∂ As␈α∞with
␈↓ ↓H␈↓many␈α
AI␈αprograms,␈α
one␈αunexpected␈α
aspect␈αof␈α
working␈α
on␈αAM␈α
was␈αthe␈α
degree␈αof␈α
precision␈αwith␈α
which
␈↓ ↓H␈↓one's ideas must be formulated.

␈↓ ↓H␈↓Everything␈αthat␈αAM␈αdoes␈αcan␈αbe␈αviewed␈αas␈αtesting␈αthe␈αunderlying␈αbody␈αof␈αheuristics.␈αGradually,␈αthis
␈↓ ↓H␈↓knowledge␈α
becomes␈α
better␈α
organized,␈α
its␈αimplications␈α
clearer.␈α
 The␈α
resultant␈α
body␈α
of␈αdetailed␈α
heuristics
␈↓ ↓H␈↓may be the germ of a more e≠cient programme for educating math students than the current dogma.␈↓	7␈↓

␈↓ ↓H␈↓But␈α
perhaps␈αthe␈α
most␈α
exciting␈αprospect␈α
opened␈α
up␈αby␈α
AM␈αis␈α
that␈α
of␈αexperimentation:␈α
one␈α
can␈αvary
␈↓ ↓H␈↓the␈α∩concepts␈α∩AM␈α∩starts␈α∩with,␈α⊃vary␈α∩the␈α∩heurisitics␈α∩available,␈α∩etc.,␈α⊃and␈α∩study␈α∩the␈α∩e≥ects␈α∩on␈α⊃AM's
␈↓ ↓H␈↓behavior.␈α∞ The␈α∞system␈α∞is␈α
just␈α∞beginning␈α∞to␈α∞run␈α∞well␈α
enough␈α∞to␈α∞make␈α∞such␈α∞experiments␈α
worthwhile.
␈↓ ↓H␈↓AM is a dissertation project ␈↓βin progress␈↓; few conclusions have been drawn yet.




␈↓ ↓H␈↓¬␈↓ ↓x********************************************************************************************************

␈↓ ↓H␈↓¬␈↓#
1␈↓#␈αThe␈αother␈α
extreme,␈αnumbers␈αwith␈α
a␈αMAXIMAL␈αnumber␈αof␈α
factors,␈αwas␈αalso␈α
proposed␈αby␈αAM␈α
as␈αworth
␈↓ ↓H␈↓¬␈↓ ↓xinvestigating.␈α∞ This␈α∞led␈α∞AM␈α∞to␈α∞many␈α∞interesting␈α∞questions;␈α∞the␈α∞only␈α∞"new-to-Mankind"␈α∞mathematical
␈↓ ↓H␈↓¬␈↓ ↓xresult␈αso␈αfar␈αis␈αthat␈αnumbers␈αof␈αthe␈αfollowing␈αform␈αare␈αall␈αmaximally-divisible:␈αp␈↓
1␈↓¬␈↓	a1␈↓¬␈αp␈↓
2␈↓¬␈↓	a2␈↓¬␈αp␈↓
3␈↓¬␈↓	a3␈↓¬...␈α p␈↓
k␈↓¬␈↓	ak␈↓¬,
␈↓ ↓H␈↓¬␈↓ ↓xwhere␈α
the␈αp␈↓
i␈↓¬'s␈α
are␈αthe␈α
first␈αk␈α
consecutive␈αprimes,␈α
and␈α
the␈αexponents␈α
a␈↓
i␈↓¬␈αdecrease␈α
with␈αi,␈α
and␈αthe␈α
ratio
␈↓ ↓H␈↓¬␈↓ ↓xof␈α(a␈↓
i␈↓¬+1)/(a␈↓
j␈↓¬+1)␈αis␈αapproximately␈α(as␈αclosely␈αas␈αis␈αpossibe␈αfor␈αintegers)␈αlog(p␈↓
j␈↓¬)/log(p␈↓
i␈↓¬).␈αFor␈αexample,␈αa
␈↓ ↓H␈↓¬␈↓ ↓xtypical␈α&divisor-rich␈α&number␈α&is␈α'n=2␈↓	8␈↓¬3␈↓	5␈↓¬5␈↓	3␈↓¬7␈↓	2␈↓¬11␈↓	2␈↓¬13␈↓	1␈↓¬17␈↓	1␈↓¬19␈↓	1␈↓¬23␈↓	1␈↓¬29␈↓	1␈↓¬31␈↓	1␈↓¬37␈↓	1␈↓¬41␈↓	1␈↓¬43␈↓	1␈↓¬47␈↓	1␈↓¬53␈↓	1␈↓¬.␈α& The
␈↓ ↓H␈↓¬␈↓ ↓xprogression␈α
of␈α
its␈α
exponents+1␈α
(9␈α6␈α
4␈α
3␈α
3␈α
2␈α
2␈α2␈α
2␈α
2␈α
2␈α
2␈α2␈α
2␈α
2␈α
2)␈α
is␈α
about␈αas␈α
close␈α
as␈α
one␈α
can␈αget␈α
to
␈↓ ↓H␈↓¬␈↓ ↓xsatisfying␈αthe␈α"logarithm"␈αconstraint.␈α This␈αnumber␈αn␈αhas␈α3,981,312␈αdivisors.␈αThe␈α"AM␈αConjecture"␈αis
␈↓ ↓H␈↓¬␈↓ ↓xthat␈α∞no␈α∞number␈α∞smaller␈α∞than␈α∞n␈α∞has␈α∞that␈α∞many␈α∞divisors.␈α∞By␈α∞the␈α∞way,␈α∞this␈α∞n␈α∞equals␈α
25,608,675,584.
␈↓ ↓H␈↓¬␈↓ ↓xThe␈α
empirical␈α
data␈α
gathered␈α
indicated␈α
the␈α∞general␈α
character␈α
of␈α
the␈α
declining␈α
exponents␈α
to␈α∞AM,␈α
but
␈↓ ↓H␈↓¬␈↓ ↓xthe␈αprecise␈αformulation␈αof␈αthe␈α"logs"␈αconstraint␈αwas␈αbeyond␈αAM,␈αsince␈αit␈αdoesn't␈αknow␈αabout␈αlogs.␈αIn
␈↓ ↓H␈↓¬␈↓ ↓xfact,␈αthis␈αwas␈αthe␈αwarped␈αway␈αin␈αwhich␈αAM␈αfirst␈αdefined␈αa␈αconcept␈αlike␈αthe␈αone␈αwe␈αcall␈α"logarithm".
␈↓ ↓H␈↓¬␈↓ ↓xThe␈αproof␈αof␈αthe␈αconjecture␈αinvolves␈αcalculus,␈αand␈αfor␈αthat␈αreason␈αwas␈αalso␈αfar␈αbeyond␈αthe␈αabilities
␈↓ ↓H␈↓¬␈↓ ↓xof␈αAM.␈αThis␈α
example␈αshows␈αthe␈αusefulness␈α
of␈αa␈αmechanical␈α
"co-researcher",␈αthe␈αeffectiveness␈αof␈α
AM-
␈↓ ↓H␈↓¬␈↓ ↓x-human cooperation.

␈↓ ↓H␈↓¬␈↓#
2␈↓#␈α∞Incidentally,␈α∞these␈α∞basic␈α∞concepts␈α∞include␈α∞the␈α∞operators␈α∞which␈α∞enlarge␈α∞the␈α∞space␈α∂(e.g.,␈α∞COMPOSE-2-
␈↓ ↓H␈↓¬␈↓ ↓xRELATIONS␈αis␈αboth␈αa␈αconcept␈αin␈αits␈αown␈αright␈αand␈αa␈αway␈αto␈αgenerate␈αnew␈αones).␈α An␈αimportant␈αkind
␈↓ ↓H␈↓¬␈↓ ↓xof␈α∞heuristic␈α∞knowledge␈α∞that␈α∂AM␈α∞possesses␈α∞is␈α∞how␈α∞to␈α∂create␈α∞new␈α∞heuristics␈α∞to␈α∞associate␈α∂with␈α∞new
␈↓ ↓H␈↓¬␈↓ ↓xconcepts␈α∪it␈α∩defines.␈α∪ For␈α∪example,␈α∩AM␈α∪has␈α∩a␈α∪rule␈α∪which␈α∩says␈α∪that␈α∩whenever␈α∪the␈α∪"looking␈α∩for
␈↓ ↓H␈↓¬␈↓ ↓xextremals"␈αheuristic␈αis␈α
used␈α("in␈αa␈αsituation␈α
f:A→B,␈αgiven␈αan␈αextremal␈α
bεB,␈αconsider␈αthe␈α
subset␈αA'=f␈↓	-
␈↓ ↓H␈↓	␈↓ ↓x1␈↓¬(b)"␈α),␈αthen␈αAM␈αshould␈αconstruct␈αa␈αheuristic␈αlike␈αthe␈αfollowing␈αand␈αtack␈αit␈αonto␈αthe␈αnew␈αconcept␈αA':
␈↓ ↓H␈↓¬␈↓ ↓x"A'␈α∩is␈α∪a␈α∩useful␈α∩space␈α∪upon␈α∩which␈α∩to␈α∪test␈α∩any␈α∩conjectures␈α∪involving␈α∩f␈α∩or␈α∪f␈↓	-1␈↓¬".␈α∩In␈α∩the␈α∪case␈α∩of
␈↓ ↓H␈↓¬␈↓ ↓xf="number-of-factors-of",␈αA␈αand␈αB␈α=␈αNatural␈αnumbers,␈αb=2,␈αthis␈αrule␈αsays␈αto␈αconstruct␈α
the␈αheuristic
␈↓ ↓H␈↓¬␈↓ ↓x"Primes␈α∂are␈α∂valuable␈α∂natural␈α∂numbers␈α∂upon␈α∂which␈α∂to␈α∂test␈α∂conjectures␈α∂involving␈α⊂multiplication␈α∂and
␈↓ ↓H␈↓¬␈↓ ↓xfactorization".␈α
This␈α∞heuristic␈α
is␈α
then␈α∞associated␈α
with␈α∞the␈α
the␈α
newly-created␈α∞Primes␈α
concept.␈α∞So␈α
the
␈↓ ↓H␈↓¬␈↓ ↓xonly␈αheuristics␈αput␈αin␈αby␈α
hand␈αpertain␈αto␈αinitially-supplied␈αconcepts.␈α
All␈αthe␈αnew␈αheuristics␈αshould␈α
be
␈↓ ↓H␈↓¬␈↓ ↓xproducable␈α∪from␈α∩those␈α∪by␈α∩AM.␈α∪In␈α∩reality,␈α∪this␈α∩is␈α∪not␈α∩always␈α∪true;␈α∩vast␈α∪improvements␈α∪in␈α∩AM's
␈↓ ↓H␈↓¬␈↓ ↓xperformance␈αare␈αseen␈αif␈αa␈αhuman␈αuser␈αoccasionally␈αsupplies␈αAM␈αwith␈αan␈αunperceived␈αheuristic.␈α Note
␈↓ ↓H␈↓¬␈↓ ↓xagain the synergetic character of the user--system dialogue.

␈↓ ↓H␈↓¬␈↓#
3␈↓# Typically, "new" means new to AM, not to Mankind, and "worthwhile" can only be judged in hindsight.

␈↓ ↓H␈↓¬␈↓#
4␈↓# Lenat, Douglas, ␈↓&BEINGS: Knowledge as Interacting Experts␈↓)αβ, 4th IJCAI, 1975, pp. 126-133.

␈↓ ↓H␈↓¬␈↓#
5␈↓#␈α∞In␈α
fact,␈α∞after␈α
AM␈α∞attempts␈α
to␈α∞find␈α
examples␈α∞of␈α
SET-EQUALITY,␈α∞so␈α
few␈α∞are␈α
found␈α∞that␈α
AM␈α∞decides␈α
to
␈↓ ↓H␈↓¬␈↓ ↓xgeneralize␈αthat␈αpredicate.␈α The␈αresult␈αis␈αthe␈αpredicate␈αwhich␈αmeans␈α"Has-the-same-length-as"␈α--␈αi.e.,
␈↓ ↓H␈↓¬␈↓ ↓xthe rudiments of Cardinality.

␈↓ ↓H␈↓¬␈↓#
6␈↓#␈α⊃Due␈α⊃to␈α⊃the␈α⊃firm␈α⊃base␈α⊃of␈α⊃preliminary␈α⊃concepts␈α⊃which␈α⊃AM␈α⊃developed,␈α⊃this␈α⊃relationship␈α⊃was␈α⊃almost
␈↓ ↓H␈↓¬␈↓ ↓xobvious.␈α AM␈αsought␈αsome␈αpredicate␈αP␈αwhich,␈αfor␈αeach␈αn,␈αsome␈αmember␈αof␈α
FACTORS-OF(n)␈αsatisfied.
␈↓ ↓H␈↓¬␈↓ ↓xALL-PRIMES␈α⊃was␈α⊃such␈α⊃a␈α⊃predicate.␈α⊃ AM␈α⊃next␈α⊃constructed␈α⊃the␈α⊃relation␈α⊃which␈α⊃associates,␈α⊃to␈α⊂each
␈↓ ↓H␈↓¬␈↓ ↓xnumber␈α⊂n,␈α⊂all␈α⊂factorizations␈α⊂of␈α⊂n␈α⊂into␈α⊂primes.␈α∂ The␈α⊂full␈α⊂statement␈α⊂of␈α⊂the␈α⊂UFT␈α⊂is␈α⊂simply␈α⊂that␈α∂this
␈↓ ↓H␈↓¬␈↓ ↓xrelation is a function; i.e., it is defined and single-valued for all numbers n.

␈↓ ↓H␈↓¬␈↓#
7␈↓#␈α
Currently,␈α
the␈α
educator␈α
takes␈α
the␈α
very␈α
best␈αwork␈α
any␈α
mathematician␈α
has␈α
ever␈α
done,␈α
polishes␈α
it␈αuntil␈α
its
␈↓ ↓H␈↓¬␈↓ ↓xbrilliance␈αis␈αblinding,␈αand␈αpresents␈αit␈αto␈αthe␈αstudent␈αto␈αinduce␈αupon.␈αMany␈αindividuals␈α(e.g.,␈αKnuth␈αand
␈↓ ↓H␈↓¬␈↓ ↓xPolya)␈αhave␈αpointed␈αout␈αthis␈αblunder.␈α A␈αfew␈α(e.g.,␈αMinsky␈αand␈αPapert␈αat␈αMIT,␈αAdams␈αat␈αStanford)␈αare
␈↓ ↓H␈↓¬␈↓ ↓xexperimenting with more realistic strategies for "teaching" creativity.