FOM: ``thoughts on the future of FOM''

[Matt Insall]

  > FOMers, what are your thoughts on the future of f.o.m. over the next
  > 100 or 1000 years?

 I'm new to the list, so I hope I'm not speaking out of turn.  (Also, I will
 make my comments brief, in the interest of not overloading email inboxes
 with too many long messages on this interesting topic.)  I predict that,
 among other things, FOM will develop much more fully its ``applied''
 perspectives, in light of the development of the science of computation in
 the past century.  The more we work on fragments of set theory, and
 variations on first and second order logic, the more computer scientists
and
 computer engineers try to adapt the new ideas to their craft.  I predict
 that the ideas developed from the interaction between researchers in FOM
and
 computer scientists and computer engineers will lead to a highly
 sophisticated form of simulated intelligence.  Moreover, the ideas and
 solutions will not only flow from ``pure'' FOM to ``applied'' FOM, but in
 both directions.  This is already begun, with projects such as QED, the
 Consequences of the Axiom of Choice Project, and others.  These currently
 take the form of fairly huge searchable databases with sophisticated search
 machinery, but I expect this will change significantly in the
not-so-distant
 future.  Directions of research in ``applied'' FOM will (I predict) include

 1.  A rigourous development of automated program verification unlike
 anything seen in the past century
 2.  An extended development of the foundations of ``fuzzy'' mathematics
 (One might call this FOFM, i.e. the ``Foundations Of Fuzzy Mathematics''.)
 3.  A growing number of researchers working on ``fuzzy'' fragments of set
 theory, first-order logic, etc. (One might call this FFOM, i.e. the ``
Fuzzy
 Foundations Of Mathematics''.)
 4.  Significant developments in the foundations of and applications of
 temporal (and modal) logic;  for instance, temporal models of set theory
 which will model the FACT that (currently, at least) many practicing
 mathematicians do not consider a statement (in or out of their area of
 expertise) to be ``true'' until it has been proved, published and
 ``undergone the test of time'', in some sense.  (This type of development
 may provide a model for one aspect of Professor Friedman's recent post
where
 he asks if all statements considered by mathematicians, either up to now or
 for all time, have some heretofore undisclosed ``special property''.  Even
 if such ``special properties'' are discovered for all of mathematics up to
 the present, it may be theoretically impossible to determine an appropriate
 such property for all mathematics for all time.  The very idea sounds
 terribly self-referential to me.  But then, appropriate models of the
 resulting development of mathematics as a whole may come in the form of an
 application of temporal logic to FOM, with the ``special properties'' being
 time-dependent.)



Matt Insall
http://www.rollanet.org/~montez
montez at rollanet.org