Re: Alexander Grothendieck on `speculation'

[Gaucher Philippe <Philippe.Gaucher@pps.jussieu.fr>]

Le Mardi 14 Mars 2006 23:43, vous avez écrit :
> Jim Stasheff writes:
> > This suggests two possibilities:
> >
> > for the brave, start your own blog for speculations
> > for the timid, same input but into a file only you can access
> > until late in life and famous you can show how you had the ideas all
> > along
>
> The situation is more complicated in that what could be classed as
> speculation may get published as theorem and proof. For example,  in
> algebraic topology, sometimes proofs of continuity are omitted as if this
> was an exercise for the reader, yet the formulation of why the maps are
> continuous (if they are necessarily so) may contain a key aspect of what
> should be a complete proof. This difficulty was pointed out to me years ago
> by Eldon Dyer in relation to results on local fibration implies global
> fibration (for paracompact spaces) where he and Eilenberg felt Dold's paper
> on this
> contained the first complete proof. I have been unable to complete the
> proof in Spanier's book, even the second edition. (I sent a correction to
> Spanier as the key function in the first edition was not well defined,
> after Spanier had replied `Isn't it continuous?')   Eldon speculated (!)
> that perhaps 50% of published algebraic topology was seriously wrong!

My guess is that most of the algebraic topologists assume that the map 
they are constructing is automatically continuous since the proof will work 
for example for simplicial sets (in which there is no continuity to check). 
And this argument is wrong : because the category of general topological 
spaces is not cartesian closed while the category of simplicial sets is 
cartesian closed. And most of these proofs of continuity become possible only 
by working in a more convenient category of topological spaces.

pg.