FOM: fom: faltings,barbers...

[marker@math.uic.edu]

Steve:

I find "what can be explained to one's barber" a very poor benchmark
for testing "general intellectual interest".  I would offer the test
"could be discussed in a reasonable length of time in a way that an
intellectually curious layman would find interesting".  My idea of an
intellectually curious layman is someone who would for example
ocassionaly read Scientific American, buy a Hawkings or Penrose book,
read a science column in the New York Times, attend a general lecture
on a topic far outside their specialty.... (I have made this sound
more science oriented than I wanted but those are the examples that
come most easily to mind).

Godel's theorem is an example that I expect we all agree is of
"general intellectual interest" but I an very sceptical that it would
pass the barber test.  I have gave a reasonably succesful lecture on
Godel's theorem to a group of bright freshman and sophmores with no
particular mathematical background.

I believe that I could give an equally succesful lecture explaining
the following fundamental mathematical insight (I have called this
"foundational" in the past, but I am quite willing for purposes of our
discussion to reserve the term "foundational" for more systematic
approaches [ I am not always this easy--I have yet to use "c.e." in a
sentence :--) ])
     TO UNDERSTAND BASIC PROBLEMS ABOUT SOLUTIONS OF POLYNOMIAL  
EQUATIONS IN THE RATIONALS OR INTEGERS YOU NEED TO UNDERSTAND  
SOMETHING ABOUT THE GEOMETRY OF THE SOLUTION SPACE.
    To illustrate this (by say Falting's theorem) one need only
understand the following concepts which we expect our students to have
at least seen in highschool: polynomial equations, complex numbers,
solution sets to equations 

Clearly in such a talk one would have to explain the usefulness of
looking at complex solutions instead of just real solutions and one
would have to introduce solutions at infinity.  But having done that
you could then state a beautiful and very deep theorem which
illustrates this fundamental insight.

Dave Marker

--PS: I believe that the benchmark Steve suggested was in fact Lou's
barber--who as far as I can tell might be Russel's "present king of
France" but I will leave this debate to others.  :-)