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

Ayan amah8857 at
Thu Jan 4 09:24:23 EST 2001

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

>(1) A question:
>Matthew Frank <mfrank at> 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!


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