Re: WHY ARE WE CONCERNED? I

[Vaughan Pratt <pratt@cs.stanford.edu>]

jim stasheff wrote:

> now for a mathematical subject a math proof is sometimes but not always
> necessary

Absolutely.  I would add publication date as a factor here.  As an
example, a few decades ago an elementary exposition of the Fundamental
Theorem of Algebra would not be expected to include an elementary proof
since the extant proofs were either lengthy arguments or nonelementary
appeals to the minimum modulus principle, properties of holomorphic
functions such as Liouville's theorem, or other results the reader would
be unlikely to be on top of.  The dominant belief was that the only
short proofs were nonelementary ones.

But for an audience aware only that z^i for any nonnegative integer i
maps circles at the origin to i-fold circles of radius r^i at the
origin, an entirely elementary notion, an expositor today would be
morally obligated to include a full proof since there is hardly anything
left to explain.  The polynomial a_d z^d + ... + a_0 maps little circles
to the neighborhood of a_0 and big circles to a loop tending to a very
big d-fold circle of radius a_d r^d, whence the smoothly growing image,
under the polynomial, of a smoothly growing circle is obliged to cross
the origin at some stage.  Still a topological argument, but now an
entirely elementary one.

Except, that is, for the theorem that a loop wound d times around the
hole in the punctured plane cannot be continuously retracted to a point,
which was tacitly smuggled in there.   But that statement is less
intimidating than anything based on holomorphic functions.

This slick proof seems only to have emerged in the past couple of
decades.  It is an interesting commentary on mathematics that it took
this long for people to come up with an argument "for the rest of us."
Maybe some people "knew" it all along, but in that case they were
keeping pretty quiet about it.

Vaughan Pratt