FOM: Some reflections on Rota

[Walter Whiteley]

The recent death of Gian-Carlo Rota (my advisor)
sent me to rereading some of his writings.  
Two small excepts from an article in the Math 
Intelligencer, Winter 1999, seemed at least tangentially
relevant to the discussions of mathematical practice
and foundations.


"Hlibert's theorem on finite generation of the ring of
invariants can be recast in the language of umbrae
and can be given a simple combinatorial proof that 
dispenses with the Hilbert Basis Theorem"


"  ...   In mathematics, it is extremely difficult to tell 
the truth.  The formal exposition of a mathematical theory 
does not tell the whole truth.  The truth of a mathematical
theory is more likely to be grasped while we listen
to a casual remark made by some expert that gives
away some hidden motivation, when we finally pin down
the typical examples or when we discover what the 
real problems are that were stored behind the 
showcase problems.  Philosophers and psychiatrists
should explain why it is that we mathematicians
are in the habit of systematically erasing our
footsteps.  Scientists have always looked askance
at this strange habit of mathematicians, which 
has changed little from Pythagoras to our day.  "


Beyond describing what mathematicians COULD
be doing IF they followed the constraints of FOM,
I do think there is an interest in describing
(giving foundational analysis for) what careful
mathematicians DO.  As I read the exchanges
between Simpson and McLarty, I see that gap
between the two descriptions of homological
algebra, cohomology theory:  what could be done - 
and - what is done as people practice the discipline.  

I note that, as a school child, I did well in 'geometry' 
- but basically found that it was an introduction to logic.  
(Unlike high school algebra which was not logicial - 
even when I was 'taught' induction.)  Much later (after 
my thesis on the 'foundations' of analytic geometry 
- i.e. invariant theory),  I learned I ALSO enjoyed geometry,
when I finally learned some of the 'truth' of geometry,
in Rota's sense.  

The effort to understand and refine what
mathematicians (e.g geometers) DO is behind 
some of the interest (my interest) in diagrammatic 
reasoning, and developing parts of the "foundations" for 
those practices which are widespread (and to be 
encouraged) in the practice of mathematics, 
in the teaching of mathematics, and in the learning
of mathematics.

Walter Whiteley
York University