COMMENT ⊗   VALID 00015 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00003 00002	.NSECP(Overview)
C00004 00003	.SSEC(Abstract of this Thesis)
C00006 00004	.SSEC(Three-page Summary of the Project)
C00009 00005	. SSSEC(Detour: Analysis of a discovery)
C00013 00006	. SSSEC(What AM does: Syntheses of discoveries)
C00021 00007	. SSSEC(Results)
C00027 00008	. SSSEC(Conclusions)
C00029 00009	.SSEC(Ways of viewing AM as some common process)
C00031 00010	. SSSEC(AM as Hill-climbing)
C00035 00011	. SSSEC(AM as Heuristic Search)
C00047 00012	. SSSEC(AM as a Mathematician)
C00056 00013	.SSEC(Fifteen-page Summary of the entire project)
C00058 00014	.SSEC(Guide to reading the remainder of the thesis)
C00060 00015	. SSSEC(Varied Readership of this thesis)
C00065 ENDMK
C⊗;
.NSECP(Overview)

.SSEC(Abstract of this Thesis)

A program,  called  "AM", is  described which  models  one aspect  of
elementary  mathematics research: developing new  concepts under the
guidance of a large body of heuristic rules.

The local heuristics  communicate via an  agenda mechanism, a  global
list of tasks for the system to  perform and reasons why each task is
plausible.  A single task might direct AM to define a new concept, or
to explore  some facet  of an existing  concept, or  to examine  some
empirical  data  for  regularities,  etc.   Repeatedly,  the  program
selects from the agenda the task having the best supporting  reasons,
and then executes it.

Each concept is  an active, structured  knowledge module.   A hundred
very   incomplete   modules   are   initially   provided,  each   one
corresponding to an elementary  set-theoretic concept (e.g.,  union).
This  provides a  definite but  immense  "space" which  AM begins  to
explore.   AM  extends its  knowledge base,  ultimately rediscovering
hundreds  of common  concepts  (e.g.,  numbers) and  theorems  (e.g.,
unique factorization).

This approach to  plausible inference contains some unexpected powers
and limitations.

.SSEC(Three-page Summary of the Project)

<<Edit this, making it follow the general outline of the thesis>

Scientists  often face  the difficult  task  of formulating  research
problems which  must be soluble yet nontrivial.   In any given branch
of science, it is usually  easier to tackle a specific given  problem
than  to   propose  interesting   yet  managable  new   questions  to
investigate.   For example, contrast ⊗4solving⊗* the Missionaries and
Cannibals problem with  the more ill-defined  reasoning which led  to
⊗4inventing⊗* it.

This   thesis  is  concerned   with  creative  theory   formation  in
mathematics: how to  propose interesting new  concepts and  plausible
hypotheses connecting them.  The experimental  vehicle of my research
is  a computer program called  ⊗2AM⊗*$$ The original  meaning of this
mnemonic has been abandoned.  As Exodus states: I ↓_AM_↓ that  I
↓_AM_↓.  The author cannot imagine how the rumor was started that
"AM" stands for "Apprentice Mathematician".
$ which  carries  out  some of  the  activities involved  in
mathematical  research:  noticing simple  relationships  in empirical
data, formulating  new  definitions out  of  existing ones,  deciding
carefully what  to explore next, evaluating the  overall worth of new
concepts.

. SSSEC(Detour: Analysis of a discovery)

Before  discussing  how  to ⊗4synthesize⊗*  a  new  theory,  consider
briefly how to ⊗4analyze⊗* one, how to construct a plausible chain of
reasoning which terminates in a given discovery.  One can do  this by
working  backwards,  by reducing  the  creative  act to  simpler  and
simpler  creative acts.   For example, consider the  concept of prime
numbers.  How might  one be led to  define such a notion?  Notice the
following plausible strategy:

.ONCE INDENT 9,9,9 SELECT 6

"If f is  a function which transforms elements of  A into elements of
B, and B is ordered, then consider just those members of A which  are
transformed  into  ⊗4extremal⊗*  elements of  B.    This  set  is  an
interesting subset of A."

When  f(x) means "divisors  of x", and  the ordering  is "by length",
this heuristic says to consider those numbers which have  a minimal$$
The other extreme, numbers with a MAXIMAL number of factors, was also
proposed  by  AM  as  worth  investigating.    This  led  AM to  many
interesting questions. See Appendix {[2]MAXDIV}.  $ number of factors
-- that is,  the primes.  So this rule  actually ⊗4reduces⊗* our task
from "proposing the concept of prime numbers" to the more  elementary
problems   of   "discovering   ordering-by-length"   and   "inventing
divisors-of".

But  suppose we  know this  general rule: ⊗6"If  f is  an interesting
function, consider its inverse."⊗* It reduces the task of discovering
divisors-of to the simpler  task of discovering multiplication$$ Plus
noticing  that  multiplication  is  associative  and  commutative. $.
Eventually, this task reduces to the discovery of very basic notions,
like substitution,  set-union, and equality.  To  explain how a given
researcher might have  made a  given discovery, such  an analysis  is
continued  until that  inductive  task  is reduced  to  "discovering"
notions which the  researcher already knew, which were his conceptual
primitives.

. SSSEC(What AM does: Syntheses of discoveries)

Suppose a  large collection  of these  heuristic strategies has been  assembled
(e.g.,  by analyzing a great  many discoveries, and  writing down new
heuristic rules  whenever  necessary).    Instead of  using  them  to
⊗4explain⊗*  how a  given idea  might have  evolved, one  can imagine
starting from  a basic core of knowledge and "running" the heuristics
to ⊗4generate⊗* new concepts.

Such syntheses are precisely what AM does.  The program consists of a
large  corpus of  primitive mathematical  concepts, each  with  a few
associated heuristics$$  Situation/action  rules  which  function  as
local "plausible move generators".  Some suggest tasks for the system
to carry  out, some suggest ways of satisfying  a given task, etc. $.
AM's activities  all serve to  expand AM  itself, to  enlarge upon  a
given body of  mathematical knowledge.  To cope with  the enormity of
the  potential "search  space"  involved, AM  uses its  heuristics as
judgmental  criteria  to guide  development  in  the  most  promising
direction.  It appears that the process of inventing worthwhile new$$
Typically, "new" means new  to AM, not  to Mankind; and  "worthwhile"
can  only  be  judged  in  hindsight.    $  concepts  can  be  guided
successfully using a collection of a few hundred such heuristics.

Each  concept is represented as  a frame-like data  structure with 25
different facets or slots.  The types of facets include:  ⊗6Examples,
Definitions,      Generalizations,      Domain/Range,      Analogies,
Interestingness,⊗*  and many  others.   The ⊗6Beings⊗* representation
provides  a convenient  scheme  for  organizing the  heuristics;  for
example, the  following strategy fits into the  ⊗4Examples⊗* facet of
the  ⊗4Predicate⊗*  concept:  ⊗6"If,  empirically,  10  times  as  many
elements ⊗4fail⊗*  some  predicate P,  as ⊗4satisfy⊗*  it, then  some
⊗4generalization⊗* (weakened  version) of P might be more interesting
than P"⊗1.    AM considers  this suggestion  after  trying to  fill  in
examples  of any  predicate$$  In  fact, after  AM  attempts to  find
examples  of  SET-EQUALITY,  so  few are  found  that  AM  decides to
generalize that predicate.  The result is a new predicate which means
"Has-the-same-length-as" -- i.e., a rudimentary precursor to Numbers.
$.

AM is initially given a collection of 115 core concepts, with only  a
few facets filled in for  each.  Its sole activity is  to choose some
facet  of some  concept, and  fill in  that particular  slot.   In so
doing,  new notions  will  often  emerge.    Uninteresting  ones  are
forgotten, mildly interesting ones are kept  as parts of one facet of
one  concept,  and very  interesting  ones are  granted  full concept
status. Such  new ⊗6Beings⊗* have  dozens of  blank parts, hence  the
space  of possible actions  (blank slots  to fill in)  grows rapidly.
The same  heuristics are  used  both to  suggest new  directions  for
investigation, and to limit attention: both to sprout and to prune.

. SSSEC(Results)

The particular  mathematical domains in  which AM operates  depend on
the  choice of initial concepts.   Currently, AM  begins with nothing
but a scanty  knowledge of  concepts which Piaget  might describe  as
⊗4prenumerical⊗*:  Sets, substitution,  operations, equality,  and so
on.     In  particular,   AM  is  not  told   anything  about  proof,
single-valued functions,  or numbers.   With this  basis, AM  quickly
discovered$$ "Discovering" a concept  means that (1) AM recognized it
as a distinguished entity (e.g.,  by formulating its definition)  and
also (2) AM decided it was worth investigating (either because of the
interesting way  it was formed, or  because of surprising preliminary
empirical results). $ elementary numerical concepts (corresponding to
those we  refer to as  natural numbers, multiplication,  factors, and
primes)  and  wandered  around in  the  domain  of  elementary number
theory.    Although  it  was  never  able  to  ⊗4prove⊗*  the  unique
factorization theorem, AM actually did ⊗4conjecture⊗* it.

AM was  not able to discover any  "new-to-Mankind" mathematics purely
on its  own,  but  ⊗4has⊗*  discovered  several  interesting  notions
hitherto unknown to the author. A couple genuinely bits of mathematics have been
⊗4inspired⊗* by AM.↑1 A synergetic AM--human combination can sometimes
produce better research than either could alone.$$ This is supported by
Gelernter's experiences with his geometry program:
While lecturing about how it might prove a certain theorem
about isosceles triangles,
he came up with a
new, cute proof. Similarly, Guard and Eastman 
noticed an intermediate result of
their SAM resolution theorem prover, 
and wisely interpreted it as a nontrivial result in lattice
theory (now known as SAM's lemma). $

Everything that AM does can be viewed as testing the  underlying body
of heuristics.   Gradually, this knowledge  becomes better organized,
its implications  clearer.  The resultant body of detailed heuristics
may be  the germ  of a more  efficient programme  for educating  math
students than  the current dogma$$ Currently, the  educator takes the
very best work any mathematician has ever done, polishes it until its
brilliance is  blinding, and  presents it  to the  student to  induce
upon. Many individuals  (e.g., Knuth and Polya) have pointed out this
blunder.   A  few  (e.g.,  Papert  at MIT,  Adams  at  Stanford)  are
experimenting   with  more   realistic   strategies  for   "teaching"
creativity.  $.

Another   benefit   of   actually   constructing   AM   is   that  of
⊗4experimentation⊗*: one can vary  the concepts AM starts  with, vary
the  heuristics  available,  etc.,  and  study the  effects  on  AM's
behavior.  Several  such experiments  were performed.   One  involved
adding a couple dozen new concepts from an entirely new domain: plane
geometry.   AM busied itself exploring elementary geometric concepts,
and was almost as  productive there as in  its original domain.   New
concepts were defined, and new conjectures formulated$$ A cute result
was  obtained   there:  any  angle  (between  0  and  180↑o)  can  be
approximated to within  one degree,  as the sum  of two angles  drawn
from the set α{0↑o, 1↑o, 2↑o, 3↑o, 5↑o, 7↑o, 11↑o,..., 179↑oα}, i.e.,
the set of angles of prime size.  $. Other experiments indicated that
AM was  more  robust than  anticipated; it  withstood  many kinds  of
"de-tuning".   Others demonstrated  the tremendous impact  that a few
key concepts (e.g.,  Equality) had  on AM's behavior.   Several  more
experiments have been planned for the near future.

. SSSEC(Conclusions)

AM is forced to judge ⊗4a priori⊗* the value  of each new concept, to
lose interest quickly  in concepts which aren't going to develop into
anything.  Often, such judgments can only be based on hindsight.  For
similar reasons,  AM has difficulty formulating  new heuristics which
are  relevant to the new  concepts it creates.   Heuristics are often
merely  compiled  hindsight.   While  AM's  "approach"  to  empirical
research may be used in other scientific domains, the main limitation
(reliance on hindsight) will probably  recur.  This prevents AM  from
progressing indefinitely far on its own.

This ultimate limitation  was reached. AM's performace  degraded more
and  more as  it progressed  further  away from  its initial  base of
concepts.   Nevertheless, AM  demonstrated that  selected aspects  of
creative  discovery in  elementary  mathematics  could be  adequately
represented  as a heuristic search process.   Actually constructing a
computer model of this activity has provided  an experimental vehicle
for studying the dynamics of plausible empirical inference.

.SSEC(Ways of viewing AM as some common process)

This section will provide  a few metaphors: some hints  for squeezing
AM  into paradigms  with which  the  reader might  be familiar.   For
example, the existence of  heuristics in AM  is quite similar to  the
presence  of  domain-specific   information  in  any  knowledge-based
system.

Consider   assumptions,  axioms,  definitions,  and  theorems  to  be
syntactic rules  for  the language  that  we call  Mathematics.  Thus
theorem-proving, and  the whole of textbook mathematics,  is a purely
syntactic process.  Then the heuristic rules used by a  mathematician
(and by  AM) would  correspond to  the semantic knowledge  associated
with these more formal methods.

Just   as   one   can   upgrade   natural-language-understanding   by
incorporating semantic  knowledge, AM is  only as  successful as  the
heuristics it knows.

Three more ways  of "viewing" AM as something else  will be provided:
(i) AM  as a hill-climber, (ii) AM as a heuristic search program, and
(iii) AM as a mathematician.

. SSSEC(AM as Hill-climbing)

Let's draw an analogy between developing mathematics and the familiar
process of  hill-climbing.  We may visualize  AM as exploring a space
using an evaluation function which imparts to it a topography.

Consider AM's  core of  very simple  knowledge.   By compounding  the
known concepts  and methods, AM  can extend this foundation  a little
wherever  it  wishes.   The  incredible  variety  of  alternatives to
investigate includes  all known mathematics,  much trivia,  countless
deadends, and so  on.  The only "successful" paths  near the core are
the narrow ribbons  of known  mathematics (perhaps with  a few  other
undiscovered slivers).

How  can  AM walk  through  this  immense  space, with  any  hope  of
following   the   few,   slender   branches  of   already-established
mathematics (or  some equally  successful new  fields)?   AM must  do
hill-climbing: As new concepts are  formed, decide how promising they
are,  always explore the  currently most-promising new  concept.  The
evaluation function  is quite  nontrivial, and this  research may  be
viewed  as  an  attempt  to  study  and  explain  and  duplicate  the
judgmental criteria  people  employ.    Attempts  at  codifying  such
"mysterious"  emotive  forces  as   intuition,  aesthetics,  utility,
richness,  interestingness, relevance...  indicated  that a large but
not unmanagable collection of heuristic rules should suffice.

The important visualization  to make is  that with proper  evaluation
criteria, AM's  planar mass  of interrelated concepts  is transformed
into  a  breath-taking relief  map:  the known  lines  of development
become mountain ranges, soaring above the vast flat  plains of trivia
and inconsistency below.

Occasionally an  isolated hill is  discovered near the  core;$$ E.g.,
Conway's numbers,  as  described  in Knuth's  ↓_Surreal  Numbers_↓  $
certainly whole  ranges lie undiscovered  for long periods  of time$$
E.g.,  non-Euclidean  geometries $,  and  the terrrain  far  from the
initial core is not yet explored at all.

. SSSEC(AM as Heuristic Search)

As the title of this section -- and this thesis -- proclaims, AM is a
kind  of  "heuristic search"  program.   That  must mean  that  AM is
exploring  a  particular  "space,"  using  some  informal  evaluation
criteria to guide it.

The flavor  of search  which is  used here  is that  of progressively
enlarging  a tree. Heuristics  are used  to decide which  node of the
tree to expand next, and to produce from that  node a few interesting
successor nodes. To do mathematical research well, I claim that it is
necessary and  sufficent  to  have good  methods  for  proposing  new
concepts from  existing ones, and  for deciding how  interesting each
candidate (concept, node) is.

AM explores mathematics  by selectively enlarging itself: AM ⊗4is⊗* a
body of mathematical knowledge (concepts, plus the wisdom to use them
effectively).   To see this, we  must explain what the  nodes of AM's
search  space  are,  what  the  operators  or  links  are, where  the
heuristic  information  comes into  play,  and  what  the  evaluation
function is.

AM's  space can  be  considered to  consist  of all  nodes which  are
consistent, partially-filled-in concepts. That is, a primitive "legal
move" for AM would be  to (i) enlarge some facet of  some concept, or
(ii)  create a new, partially-complete  concept. Consider momentarily
the size of this space.  Since there is no constraint on what the new
concepts  can  be, and  no  informal  knowledge for  quickly  finding
entries  for a desired  facet, a blind "legal-move"  program would go
nowhere --  slowly!   One shouldn't  even  call the  activity such  a
program would be doing "math research."

The heuristic  rules are used as little  "plausible move generators".
They suggest what  facet of what  concept to enlarge  next, and  they
suggest specific new concepts to create. The only activities which AM
will  consider doing  are those  which have  been motivated  for some
specific good reason. The validity  of that last statement of  course
depends  on  the validity  of  the  heuristic  rules.   This  is  the
programmer's responsibility.

AM  has  a definite  algorithm  for rating  the nodes  of  its space.
Namely,  the   heuristic   rules   provide  enough   information   to
meaningfully  order the  tasks on  the agenda  list.   Yet AM  has no
specific goal criteria: it  can't quit just  because a dynamite  task
has been proposed. AM goes on forever$$ Technically, forever is about
100,000 list cells and a couple cpu hours. $.

Consider  Nilsson's  description  of  depth-first  searching, and  of
breadth-first searching. He has us  maintain a list of "open"  nodes.
Repeatedly, he  plucks the top  one and  expands it. In  the process,
some  new  nodes may  be  added  to the  Open  list. In  the  case of
depth-first searching, they  are added at the  top; the next node  to
expand is the one most  recently created; the Open-list is being used
as a push-down stack.  For breadth-first search, new nodes are  added
at the bottom;  they aren't expanded  until all the older  nodes have
been;  the Open-list is  used as a  queue.  For  heuristic search, or
"best-first" search, they are evaluated in some numeric way, and then
"merged" into the already-sorted list of Open nodes.

.ONCE TURN ON "{}"

This process is  very similar to the ⊗4agenda⊗* mechanism  AM uses to
manage  its  search. This  will  be  discussed in  detail  in Chapter
{[2]AGENDA}.  The agenda is  a list of plausible tasks for AM  to do,
plus supporting reasons for each  task.  When a task is suggested for
some reason, it  is added to  the agenda.   A task  may be  suggested
several times,  for different reasons.   A  global priority value  is
assigned  to each task, based  on the combined value  of its reasons.
The control structure  of AM is  simply to select  the task with  the
highest priority,  execute it, and  select a  new one.   The "agenda"
appears  to  be a  very  well-suited  data structure  for  managing a
"best-first" search process.

Similar control  structures were  used in  Dendral$$ The  "Predictor"
part  of DENDRAL.  See  [MI4 ref.].  $, SIMULA  [reference],  and KRL
[reference].  The main difference is that in AM, symbolic reasons are
used (albeit in trivial token-like ways) to decide whether -- and how
much -- to boost the priority of a task when it is suggested again.

There are  several difficulties and anomalies in  forcing AM into the
heuristic search  paradigm.  AM's  heuristics are  used as  plausible
move generators;  if they are all  removed, AM would have  nothing to
do.   In traditional heuristic searches,  the heuristics are separate
from a "legal  move generator", hence  could all be eliminated:  they
merely help constrain a single generator.

Another anomaly  is that the operators  which AM uses  to enlarge and
explore the space  of concepts are  themselves mathematical  concepts
(e.g., some  heuristic rules  result in  the creation  of new  rules;
"Compose"  is both a  concept and an  operation which  results in new
concepts).  Thus  AM should be  viewed as a  mass of knowledge  which
enlarges  ⊗4itself⊗* repeatedly.   As  far  as I  know, all  previous
computer  programs  kept  the  information  they  "discovered"  quite
separate from the knowledge they used to make discoveries$$ Of course
this  is  typically  because the  two  kinds  of  knowledge are  very
different:  For  a  chess-player,  the  first  kind  is  "good  board
positions," and the  second is "strategies  for making a  good move."
So-called "learning  programs" typically learn one specific task, not
how to become better learners. etc. $.

Perhaps the  greatest  difference between  AM and  typical  heuristic
search procedures is  that AM has no well-defined  target concepts or
target relationships.  Rather, its "goal criteria" -- its sole aim --
is  to  maximize the  interestingness  level  of  the  activities  it
performs, the  priority ratings of the  top tasks on the  agenda.  It
doesn't  matter  precisely  which   definitions  or  conjectures   AM
discovers -- or misses -- so long as it  spends its time on plausible
tasks.  There is no fixed set of theorems that AM should discover, so
AM is  not a typical  ⊗4problem-solver⊗*. There  is no  fixed set  of
traps AM should avoid, and no winning/losing behavior, so AM is not a
typical ⊗4game-player⊗*.

For  example,  no  stigma  is attached  to  the  fact  that  AM never
discovered real  numbers$$  There are  many  "nice" things  which  AM
didn't -- and can't -- do: e.g., devising ↓_geometric_↓ concepts from
its initial simple  set-theoretic knowledge.   See the discussion  of
the limitations of AM, Section  {[2]DIFSECNUM}.{[2]DIFSSECNUM}. $; it
was  rather  surprising  that  AM  managed  to  discover  ⊗4natural⊗*
numbers!    Even  if  it  hadn't  done  that,  it  would   have  been
acceptable$$ Acceptable  to whom? Is there  really a domain-invariant
criterion  for  judging  the  quality  of  AM's  actions?    See  the
discussions in Section {[2]EVALU}.1. $ if AM had simply  gone off and
developed ideas in set theory.

. SSSEC(AM as a Mathematician)

Before diving into the innards of AM, let's  take a moment to discuss
the  totality  of the  mathematics  which  AM carried  out.    Like a
contemporary  historian  summarizing  the  work  of  the   Babylonian
mathematicians, we shan't hesitate to use current terms and criticize
by current standards.

AM   began  its  investigations  with  scanty   knowledge  of  a  few
set-theoretic concepts  (sets,  equality  of sets,  set  operations).
Most  of the obvious  set-theory relations  (e.g., de  Morgan's laws)
were eventually uncovered; since  AM never fully understood  abstract
algebra, the statement  and verification of  each of these  was quite
obscure.    AM never  derived a  formal  notion of  infinity,  but it
naively established conjectures like "a set can never be a  member of
itself", and procedures for making chains  of new sets ("insert a set
into  itself").  No sophisticated  set theory (e.g., diagonalization)
was ever done.

After this initial period of exploration, AM  decided that "equality"
was   worth  generalizing,  and   thereby  discovered   the  relation
"same-size-as".  "Natural numbers" were based on this, and soon  most
simple arithmetic operations were defined.

Since addition arose as an analog to  union, and multiplication as an
analog to cross-product,  it came as quite a surprise when AM noticed
that they  were related (namely,  N+N=2xN).   AM later  re-discovered
multiplication in three other ways: as repeated addition, as iterated
substitution$$  Take two bags A  and B. Replace each  element of A by
the bag B. Remove one level of parentheses by taking the union of all
elements of  the transfigured bag A.  Then that new bag  will have as
many elements as the product of the lengths of the two original bags.
$, and by  studying the cardinality of  power sets$$ The size  of the
set of all subsets of S is 2↑S.  Thus the power set of A∪B has length
equal to the ↓_product_↓ of the lengths of the power sets of A  and B
individually (assuming  A and B  are disjoint). $.   These operations
were  defined  in diffeent  ways,  so it  was an  unexpected  (to AM)
discovery when they all turned out to be  equivalent. These surprises
caused AM to give Times quite a high Worth rating.

Exponentiation was defined as repeated multiplication. Unfortunately,
AM never found any obvious  properties of exponentiation, hence  lost
all interest in it.

Soon after  defining multiplication, AM  investigated the  process of
multiplying a number by itself: squaring.  The inverse of this turned
out to be interesting, and led to the definition of square-root.   AM
remained   content    to   play   around   with    the   concept   of
⊗4integer⊗*-square-root. Although it defined the set of numbers which
had no square root, AM was never close to  discovering rationals, let
alone irrationals.

Raising to fourth-powers, and fourth-rooting, were discovered at this
time.  Perfect squares and perfect fourth-powers were isolated.  Many
other numeric  operations and kinds  of numbers were  isolated: Odds,
Evens, Doubling, Halving, etc.

.ONCE TURN ON "{}"

The associativity  and commutativity of multiplication indicated that
it could accept a  BAG of numbers as its  argument.  When AM  defined
the inverse  operation corresponding to Times,  this property allowed
the  definition to be: "any  ⊗4bag⊗* of numbers  whose product is x".
This   was   just   the    notion   of   factoring   a    number   x.
Minimally-factorable numbers  turned out to  be what we  call primes.
Maximally-factorable numbers were also thought to be interesting.

AM  conjectured  the  fundamental   theorem  of  arithmetic   (unique
factorization  into primes)  and  Goldbach's conjecture  (every  even
number >2 is  the sum of two primes) in a surprisingly symmetric way.
The unary representation of numbers gave way to a representation as a
bag of primes  (based on unique factorization), but  AM never thought
of  exponential  notation.   Since  the  key  concepts  of remainder,
greater-than, gcd, and  exponentiation were never mastered,  progress
in number theory was arrested.

When a  new base of  ⊗4geometric⊗* concepts was  added, AM  began finding
some  more general associations.  In place of  the strict definitions
for  the  equality  of   lines,  angles,  and  triangles,  came   new
definitions  of  concepts we  refer  to  as Parallel,  Equal-measure,
Similar,  Congruent, Translation,  Rotation, plus many  which have no
common name (e.g. the relationship of two triangles  sharing a common
angle).  A cute geometric interpretation of Goldbach's conjecture was
found$$  Given   all   angles  of   a   prime  number   of   degrees,
(0,1,2,3,5,7,11,...,179 degrees),  then any angle  between 0  and 180
degrees can be approximated (to within 1 degree) as the sum of two of
those angles. If our culture and our technology were  different, this
result  might have  been a  well-known one.  $.   Lacking a  geometry
"model" (an analogic representation like the one Gelernter employed),
AM  was  doomed  to  failure  with  respect  to  proposing  geometric
conjectures.

Similar restrictions due to poor "visualization" abilities would crop
up in topology.   The concepts of  continuity, infinity, and  measure
would have  to be  fed to  AM before  it could enter  the domains  of
analysis. More and  more drastic changes in its initial base would be
required, as the desired domain gets further and further  from simple
set theory.

.SSEC(Fifteen-page Summary of the entire project)

<<This section is not written yet. Sorry. >

<potential organization: mirror the overall organization of the thesis itself>

Include the following points on Motivation (why is this worthwhile?):

.B

	Inherent interest of getting a handle on the task (sci. creativity)
		Personal belief that discovery can be (ought to be) demystified
		Potential for learning, from the system, more about the process 
			of sci. concept formation, thy. formation, chance discovery
			(do experiments on the implementations: eg, vary AM's heurs)
	Potential usefulness of the implementations themselves (including AM)
		Aids to research; i.e., ultimately: new discoveries.
		Potential to education: like Mycin, extract heurs. and teach them
	All the usual bad reasons:
		"Look ma, no hands" + maternal drives + ego + thesis drives +... 
	Historical: 
		Need task with no specific goal, to test BEINGs ideas.
		Disenchantment with theorem-provers that plod along, in contrast
			to the processes which my model of math demands: intu, need,
	                aesth., multiple reprs, proposing vs proving, fixed task.
	
.E

.SSEC(Guide to reading the remainder of the thesis)

<<This guide is not written yet. Sorry. >

.B

	i) Overall organization of the thesis
	ii) Plans for what to read (and in what order), depending on your interests
		Plan for those interested in the AI ideas
		Plan for those interested in the systems ideas
		Plan for those interested in mathematics
	iii) Pre-requisites and how to satisfy them, for each chapter
		For those with little pure mathematics in their background
		For those with little computer science background
		For computer scientists with little contact to AI before
	    <either organized by "type" of reader, or by chapter/section>

.E

. SSSEC(Varied Readership of this thesis)

This thesis  -- and its  readers --  must come to  grips with a  very
interdisciplinary  problem.   For the reader  whose background  is in
Artificial  Intelligence,  most  of  the  system's  actions   --  the
"mathematics" it does  -- may seem inherently  uninteresting. For the
mathematician,  the  word "LISP"  signifies  nothing beyond  a speech
impediment  (to Artificial  Intelligence  types  it also  connotes  a
programming impediment). If I  don't describe "LISP" the first time I
mention it, a large fraction of potential readers will never  realize
that potential. If I ⊗4do⊗* stop to  describe LISP, the other readers
will be bored.

.ONCE TURN ON "{}"

In  an attempt not to  lose readers due to  jargon, two glossaries of
terms have  been  compiled.  Appendix  {[2]GLOS}.1  contains  capsule
descriptions of about 100  mathematical terms, ideas, notations, etc.
Appendix  {[2]GLOS}.2  renders the  analogous service  for Artificial
Intelligence jargon and computer science concepts.