FOM: The Evolution of Mathematical Theories

[Walter Whiteley]

I have had some experience with the `evolution’, `death’ and `revival’
of mathematical fields.   First, I moved on out of logic, during my
graduate days (after three advisors) in part because I could not get a
feel for the `larger questions’ - the motivations in Harvey's words,
which drove the then current research in areas like recursion theory.

 I moved to a truly dead field:  Invariant Theory.  Martin Davis has
already given his version of this `demise’ (and revival) which I
received as I was drafting this.  Let me give a slightly different
(perhaps complementary) version as it was reported to me.  There are two
articles, which I read at the time:

Charles S. Fisher, The Death of a mathematical theory: a study in the
sociology of knowledge, Archives for History of Exact Sciences III
(1966), 137-59.

Charles S. Fisher, The last of the invariant theorists, Arch. Europ.
Sociol. VIII (1967), 216-244

The `last of the invariant theorists’ was Prof. Turnbull in Edinburgh.
When he died, my advisor (Rota) visited the university and was taken to
his office and told he could box up whatever he wanted to take, because
no one there was interested in the material!
 I am not sure that the evolution to death was entirely determined by
`intellectual’ reasons.  The stories I heard included some sociological
reasons for the death of a theory.
The big people leave (Hilbert, Noether, van der Waerden …).   In this
case, the last two people had bad experiences with their advisors and
their theses, and, the story goes, consciously decided to omit invariant
theory from the `modern’ algebra text they were creating.  In addition,
at the level of motivation, I would say that the original motivation,
which to me was geometric with an algebraic overlay, had been twisted
into the purely algebraic / algorithmic questions of finding finite
generators for ideals.  Hilbert solved the existence problem (and left
some notes on the algorithmic problems which have recently been revived
by people like Bernd Sturmfels).  A story illustrates one of the issues
of `lack of motivation’.  When, around the 1880’s, one of the leaders
was asked what invariant theory was good for he reportedly answered:  it
is good for a lot of theses!
  I worked in invariant theory as the foundations of analytic geometry,
for a few years, and learned some essential skills at computation,
projective geometry  etc.  I left the field, per se, when I realized
almost nobody was interested in what I might write about many of the
questions, though I published part of my thesis then, and another part
20 years later when some people in computational algebraic geometry were
interested in results.  I also use those skills routinely in my own work
(like today when I was analyzing rigidity of frameworks in hyperbolic
space using Grassmann Algebra ? see below).

 I would propose discrete geometry, particularly of the classical sort
related to projective, hyperbolic, spherical, Euclidean etc. is another
`dead’ field.  Having the intellectual `luxury’ of a job with no
expectations of research (teaching at the rough equivalent of a junior
college), I switched into discrete applied geometry from invariant
theory.  [As Harvey indicates, my choice of fields did reduce
drastically the University positions for which I would be considered.
The fact that I evidently wanted to teach and thought teaching was
important probably reduced the options even more than my field of
research!  Sociologically, having people to work collaborate with in the
city where I lived was an essential factor in my switch ? but these
people were NOT primarily research mathematicians. ]
 As evidence that discrete geometry `died’, I would offer the fact that
it is not taught in graduate programs, nor to honours math students
except those heading for high school teaching.  [As an aside, I would
note that Mathematical Logic is also becoming a course primarily taught
as a service course for computer science students, philosophers, etc.]
When Hilbert posed his famous problems, substantial parts of three of
these were in discrete geometry.  The first one solved, by his student
Dehn, was in discrete geometry (a counter example to the
equidecomposability of polyhedra of the same volume).  In the 1970’s
when there was a conference on Hilbert's problems and new sets of
problems were proposed, none of the new problems were in discrete
geometry.  On the other hand, part of one of these problems was
apparently solved just last year:  the optimality of the standard sphere
packing in 3-space.  [The papers are still being reviewed, though a
workshop at Princeton has examined the proof and not found major
difficulties.  This is in contrast with the previous `proof’ by someone
else, also published, which contained lemmas for which there were
counterexamples!]
 As further evidence of the `death’ of discrete geometry, I would offer
what happened at U of T, home for arguably the best known geometer of
this century: Donald Coxeter.  As people from that `school’ retired,
died, etc. they have not been replaced.  Today there is one person in
finite projective planes left and he probably will not be replaced when
he retires.  U of T had so little interest in Coxeter’s work that York
(another University in Toronto) has Coxeter’s preprint/reprint
collection, and a number of his models.  We also have more geometers
working in our graduate program than U of T.
 I think one contributing factor to the `decline’ of discrete geometry
is the form of teaching of high school and service undergraduate
courses.  They often become exercises in `logic’ and mental gymnastics,
without any geometric flavour or motivation.  They are designed to teach
`what is a proof’ not `what is an interesting geometric question’.  One
reason this is (now) true is that the courses are taught by people who
themselves who never learned geometry in a coherent, motivated form.
(Often they are `logicians’ who teach geometry as a history of axiom
systems!) .  I note that I am typical of people now practicing discrete
geometry in North America: I have NOT had a geometry course since high
school!  [The other major group is those trained in Europe: Germany,
Austria, Switzerland, Italy, the former Soviet Union, etc. where there
was not the same decline in geometry.  The fact that there are such
`regional variations’ suggests important cultural and sociological
factors in what declines.]

  I now worked in discrete applied geometry for about 25 years ? driven
by real applications in structural engineering, computational geometry,
CAD etc., as well as the rich history of statics, theorems of Clerk
Maxwell and Cremona, and interesting analogies and correspondences with
fields like multivariate splines in CAGD.   These motivations related to
Harvey's comments on revival of a field.  I would assert that discrete
geometry is reviving, from sources in applications as well as some
general intellectual issues such as `visualization’, studies of
perception and cognition, etc.  To get an indication of the extent of
the `revival’ of discrete geometry, I can suggest:

Jacob Goodman and Joseph OíRourke: Handbook of Discrete and
Computational Geometry, CRC Press, 1997.

I won't go into this revival in detail, but if people are interested, I
am in the process of writing a longer paper (for an audience in
mathematics education) on the decline and rise of geometry, which should
be submitted in the fall

--
Walter Whiteley
Mathematics and Statistics
York University
whiteley at mathstat.yorku.ca
416-726-2100  ext 33971