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. SSSEC(Limitations of the Model of Math Research)

AM, like  anything else in  this world, is  constrained by a  mass of
assumptions. Most of these are "compiled" or interwoven into the very
fabric of AM,  hence can't  be tested by  experiments on AM.   AM  is
built around  a particular  model of  how mathematicians  actually go
about   doing  their   research.     This  model  was   derived  from
introspection, but can  be supported by  quotes from Polya, Kershner,
Hadamard, Skemp, and many others.  No attempt will be made to justify
any of  these  premises.    Here is  a  simplified  summary  of  that
information processing model for math theory formation:

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λλ The order in which a math textbook presents a theory is almost the
exact opposite  of the order in which  it was actually discovered and
developed.  In a text, new  definitions are stated with little or  no
motivation, and they turn out to be just the ones needed to state the
next big  theorem, whose proof then magically appears. In contrast, a
mathematician  doing   research  will   examine  some   already-known
concepts, perhaps trying to find some regularity in experimental data
involving them. The patterns he  notices are the conjectures he  must
investigate further, and these relationships directly motivate him to
make new definitions.

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λλ  Each step  the  researcher takes  while developing  a  new theory
involves choosing from  a large set of  "legal" alternatives --  that
is,  searching.   The  key to  keeping this  from  becoming a  blind,
explosive  search  is the  proper  use of  evaluation  criteria. Each
mathematician uses his own  personal heuristics to choose  the "best"
alternative available.

λλ   Non-formal   criteria   (aesthetic  interestingness,   inductive
inference from  empirical evidence,  analogy, and  utility) are  much
more  important   than   formal  deductive   methods  in   developing
mathematically   worthwhile   theories,   and   in  avoiding   barren
diversions.

λλ Progress in ⊗4any⊗*  field of mathematics demands much  non-formal
heuristic  expertise  in  ⊗4many⊗*  different  "nearby"  mathematical
fields.  So a  broad, universal  core of  knowledge must  be mastered
before any single theory can meaningfully be developed.

λλ It is sufficient (and  pragmatically necessary) to have and  use a
large  set  of  informal  heuristic  rules. These  rules  direct  the
researcher's next activities, depending  on the current situation  he
is  in.   These rules  can  be assumed  to  superimpose ideally:  the
combined  effect of several rules  is just the sum  of the individual
effects.

λλ The  necessary  heuristic rules  are  virtually  the same  in  all
branches of mathematics,  and at all levels of  sophistication.  Each
specialized  field will  have some of  its own  heuristics; those are
normally much more powerful than the general-purpose heuristics.

λλ For true understanding, the researcher should  grasp$$ 
Have access
to,  relate to,  store,  be able  to  manipulate, be  able to  answer
questions about  $  each  concept  in  several  ways:  declaratively,
abstractly, operationally,  knowing  when it  is relevant,  and as  a
bunch of examples.

λλ  Common metaphysical assumptions about  nature and science: Nature
is  fair,   uniform,   and  regular.   Coincidences   have   meaning.
Statistical  considerations are  valid when  looking at  mathematical
data.  

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<<Remove this note before final version!!>

<<Perhaps incorporate some of Iberall's metaphysics here, in final point.>