From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Are there any Intuitionists left?
Date: 29 Jul 1996 16:45:03 GMT

In article <4th8ra$6e0@bolt.Lakeheadu.Ca>,
Bob Forslund <bforslun@sky.lakeheadu.ca> wrote:
>"Vincent R. Johns" <vjohns@cliff.backbone.uoknor.edu> wrote:
>
>>I hope someone else hasn't already mentioned this, but with the
>>dates you mentioned, he should be famous for more than just his
>>Fixed-Point Theorem.  I found a short biographical citation:
>>
>>L(uitzen) E(gbertus) J(an) Brouwer 
>[...]
>
>What is the Fixed-Point Theorem?
>

Brouwer's Fixed-Point Theorem: given any continuous function  f: X -> X
where  X  is the unit disc in Euclidean space, there is a fixed point  p,
i.e., a point  with  f(p) = p. 

In 1-dimensional space this is easily proved with tools from calculus.
In higher dimensions, the usual proofs are topological, and are proofs
by contradiction (if there were no fixed points, there would be a
retraction to the boundary, which is prohibited for topological
reasons). Brouwer's Intuitionism rejected non-constructive existence
proofs, so that his philosophy would force him to dismiss his own
mathematics. 

There are also constructive proofs.

The theorem in 3-dimensional space may be presented imprecisely but
viscerally like this: no matter how you stir your coffee, at least one
molecule of it has to be returned to its original location.

dave