FOM: qualifications for judging g.i.i.; a philosophical exercise

[Stephen G Simpson]

This posting is about the question of why or how f.o.m. is of general
intellectual interest.  It seems that some people on the FOM list have
not seriously come to grips with this question.

Consider for instance the remarks of Torkel Franzen 17 Mar 1998
12:54:33:
 > Now, this enormous (and, I forgot to add, objective) general
 > intellectual interest of f.o.m. which is independent of whether
 > anybody actually takes an interest in it isn't really what people
 > have in mind ....  It doesn't make a great deal of sense to discuss
 > the general intellectual interest of something unless one makes an
 > attempt to relate it to the actual intellectual concerns of other
 > people .... what is lacking is any argument linking f.o.m. to the
 > actual intellectual concerns of non-specialists.

Franzen is saying that, in order to establish the general intellectual
interest of f.o.m., there is a need to "relate" or "link" f.o.m. to
unspecified "intellectual concerns of other people".  But which
intellectual concerns?  Is Monday night football relevant?  I don't
think so, but where do we draw the line?  Franzen's point has no
serious content.  His proposed public relations campaign will never
lead to anything of interest.

How did Franzen arrive at such an impasse?  Perhaps it's my fault for
talking about barbers way back in November.  Or perhaps it is really a
profound philosophical misunderstanding, along the lines of "general
interest = vacuousness" (Heidegger? Wittgenstein?).  I'll ponder this
some more.

In the meantime, let me try to clear up the immediate
misunderstanding, by making a few new points.

First, f.o.m. *already is* related to the concerns of the man in the
street.  This is because f.o.m. deals with the logical structure of
mathematics, and mathematics is applied to develop technology which
benefits the man in the street, whether he understands it or not.  But
note that this indirect relationship doesn't imply that f.o.m. is in
need of a public relations campaign a la Franzen.

Second, f.o.m. is a highly developed subject with a lot of great
achievements by people like Frege, Turing, G"odel, Cohen, et al, and
is also closely tied to mathematics itself, which has its own body of
problems and techniques.  Clearly nobody can expect to get a
*detailed* understanding of f.o.m. and its intellectual significance,
let alone specific f.o.m. advances, without mastering at least some of
the relevant background material.

The generality of "general intellectual interest" does not imply that
any and every ignorant lout is especially well qualified to judge such
matters by virtue of his ignorance.  Actually, the opposite is the
case: the *more* you know, the better qualified you are to judge such
issues.

What then *is* the general intellectual interest of advances in
f.o.m.?  Ultimately this is a philosophical matter, concerned with the
place of mathematics in the structure of human knowledge as a whole.
If you can gain even a little bit of new insight with respect to such
matters, and if you can correctly formulate your new insight in
appropriately broad and objective terms, then "general intellectual
interest" (g.i.i.) is an appropriate accolade.  Of course there are
degrees here, and not all f.o.m. advances are of equal g.i.i., but the
principle is clear.

In the case of Friedman's recent results on "greedy Ramsey theory", I
think an appropriate g.i.i. formulation would read something like the
following:

  A coherent body of results in finite mathematics, related to data
  structures which are familiar in computer applications, have been
  shown to be provable only by use of speculative mathematical axioms.
  These speculative axioms are strong axioms of infinity which go far
  beyond the standard mathematical axioms which have hitherto sufficed
  for the bulk of mathematical practice.  Such uses of speculative
  axioms is unprecedented.

As a useful series of philosophical exercises, let's try to give
comparable g.i.i. formulations of other well known high points of
f.o.m. research.  I have in mind advances such as Frege's invention of
the predicate calculus, G"odel's completeness and incompleteness
theorems, Turing's work on computability, the consistency and
independence of the continuum hypothesis, the large cardinal
hierarchy, the relationship between determinacy and large cardinals,
etc.

These exercises are philosophical, in the highest sense of the term,
because each of them involves a ruthless pruning away of all that is
inessential with respect to general intellectual interest.

-- Steve