Re: [HM] History in Mathematics

Gordon Fisher (
Sat, 07 Aug 1999 14:09:07

At 12:54 PM 8/7/99 -0400, Ralph A. Raimi wrote:
> While I am (or was) too young to know many details, I believe
> Wilder's notable achievements included the attraction to Michigan of
> a galaxy of topologists, including Sammy Eilenberg before the war
> (Eilenberg got out of Europe just in time), Samelson, Steenrod, and
> Raoul Bott. I don't know why he should have been "drummed out" of
> anything, since he himself taught an R.L. Moore - style course for
> undergraduates, in which he gave out a paced list of axioms and
> definitions and required the students to prove (at the blackboard,
> too, when their turn came) the theorems he also fed out gradually
> as the semester progressed. Gail Young, another Moore topologist at
> Michigan, also taught a course partly in the same manner. I don't
> remember the axiom system Wilder used as I never took his course,
> but Young's were axioms for the closed line interval as an ordered
> continuum with end points. ("non-cut-points")

Well, I guess "drummed out" is an example of a tendency I have toward
hyperbole. I do seem to recall someone, maybe Ed Moise or Gail Young, that
Moore was displeased to some degree at Wilder's attitude toward manifolds.
Moore, I think, wanted people to work out 3-dimensional manifolds in his
special geometric and axiomatic and "independent-of-history-and-traditions"
manner rather than deal with n-dimensional manifolds, and took a somewhat
dim view of algebraic methods in the study of manifolds -- homology and
homotopy, and all that.

In any case, it does seem that Moore made some effort to hold together a
coterie of Moore topologists who would devote themselves to problems he was
interested in. R H Bing was an example of a quite faithful such
topologist, I think, whereas Wilder was not faithful, nor was Gail Young,
although I think Young stayed closer to the "Moore school" than Wilder.

It seems to me that teaching a Moore style course to undergraduates would
be consistent with the differences between Moore's and Wilder's research
interests. Very likely the Moore approach, the so-called "Moore method",
to teaching from a set of axioms (preferably Moore's set in his AMS book)
without reference to or even knowledge of previous work in topology --
appealed to Wilder, as it has to many other topologists influenced by Moore.

I remember Hans Samelson well. He took over the differential geometry
course I was in when the incumbent teacher (was his name Myers?) dropped
dead from a heart attack at a Michigan football game. Also, in connection
with Wilder and Bott, I remember being told of an incident in which Bott
was looking over a dissertation directed by Wilder, a long piece which
studied at length certain kinds of manifolds. Bott is said to have burst
out laughing when he noticed that the initial axioms for the manifolds were
inconsistent, and that there were no such manifolds, except for the empty
set. I expect the writer of that dissertation did not burst out laughing
at this discovery.

Incidentally, I spent a year and a summer as a graduate student at the
University of Michigan, I think it was 1955-1956.

Gordon Fisher