FOM: human well-being; constructivism; anti-foundation

[Ayan]

At 06:34 PM 1/2/01, Andrej wrote:

>(1) A question:
>
>Matthew Frank <mfrank at math.uchicago.edu> writes:
> > Hilbert responded to Brouwer (according to Reid's biography, p. 184)
> > with a slight variation of this: "with your methods most of the
> > results of modern mathematics would have to be abandoned, and to me
> > the important thing is not to get fewer results but to get more
> > results."
>
>I've heard several times (e.g. Beeson's Foundations of Constructive
>Mathematics) that both Hilbert and Brouwer _erroneously_ expected that
>"most results of modern mathematics would have to be abandoned" in an
>intuitionistic setting. Does anyone know of _specific_ remarks or
>comments by Hilbert and/or Brouwer that would indicate more precisely
>which theorems or parts of mathematics they expected to evaporate
>under intuitionism?
>


Can't talk about Brouwer's Intuitionism, but in Bishop's mathematics 
general topology so important and interesting in classical setting is 
missing. Here is what Bishop wrote in beginning of  Chapter 3, Bishop-Bridges.
"Very little is left of general topology after that vehicle of classical 
mathematics has been taken apart and reassembled constructively".

I once heard from Bridges that this was probably a premature comment. How 
important is topology? Any comments!

--Ayan