From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: Foundational matters Was: Re: Axiom of Choice
Date: 10 Jul 2000 13:24:44 -0400
Newsgroups: sci.math
Summary: [missing]

david_ullrich@my-deja.com writes:

>  kramsay@aol.commangled (Keith Ramsay) wrote:
>>
>> Ed Nelson at Princeton has also expressed concerns with the validity
>> of induction.
>
>    So I'd heard. But I'd never heard much about him other than
>that he had curious attitudes about the integers - otoh with
>M-S I have independent reasons for regarding him as a
>competent mathematician. (This is not to say that Nelson
>isn't, just a statement about what I happen to know from
>my own experience.)

Years before he (publically; or at least, in print--I have no
idea what he might have said in the Fine Hall common room)
"expressed concerns with the validity of induction", he was
a notable analyst of a somewhat algebraic bent.  I have several 
of his books from the old yellow Princeton "Mathematical Notes" 
series, in particular _Tensor Analysis_, in which he works out
the example of the 4-dimensional configuration space M of a
(rather idealized) automobile.  "There are two distinguished 
vector fields, called Steer and Drive, on M corresponding
to the two ways in which we can change the configuraiton of
a"nd two more which he calls Slide and Rotate.  After some
calculations, "the Lie product of Steer and Drive is equal
to Slide + Rotate on" the submanifold of M where the wheels
are pointing in the same direction as the car.  "Let
us denote the Lie product of Steer and Drive by Wriggle.
Then further simple computations show that we have the
commutation relations

  [Steer, Drive]   = Wriggle,
  [Steer, Wriggle] = -Drive
  [Wriggle, Drive] = Slide,

and the commutator of Slide with Stter, Drive, and Wriggle is
zero.  Thus the four vector fields span a four dimensional 
solvalble lie algebra over R.  To get out of an extremely
tight parking spot, Wriggle is insufficient because it may
produce too much rotation.  The last commutation relation shows,
howeer, that one may get out of an arbitrarily tight
parking spot in the following way: wriggle, drive, reverse 
wriggle (this requires a cool head), reverse drive, wriggle, 
drive, ... ."

All in all a wonderful illustration of holonomy.

Lee Rudolph