[FOM] Constructivism, Geometry, and Powerset

[Vasco Brattka]

If one looks at the Intermediate Value Theorem as it is treated
in constructive mathematics as a classical mathematician (this
is what I am trying to do here), then one very meaningful
interpretation is that constructive results are uniform results.

That is the question about the Intermediate Value Theorem
is not only, whether

- there exists a zero x for any continuous function f
   that changes its sign on the unit interval,

but whether there is a continuous map

- f->x

that maps the continuous function f to one of its zeros x.

It is a plain classical fact that there is no such
continuous map in general (not even a multi-valued one).
This observation has nothing to do with constructive mathematics.
But the failure of the Intermediate Value Theorem in
constructive mathematics reflects exactly this fact.

 From this point of view, the failure of the Intermediate
Value Theorem in constructive mathematics is in excellent
correspondence with the classical mathematical intuition.

The distinction between uniform and non-uniform results
is integrated in computable analysis, which is a classical
theory (and a model of constructive analysis as classical
analysis is). What is known about the Intermediate Value
Theorem in computable analysis is

- that there is no uniform computable solution in general
   (because there is no continuous multi-valued map of
    the type f->x mentioned above),

- restricted to subclasses (such as the class of continuous
   functions whose zero sets do not contain any interval)
   there is a uniform computable solution,

- any computable map f on the unit interval that changes
   its sign has a computable zero although we cannot find
   it uniformly (the proof requires a non-constructive
   case distinction).

I believe that it is inappropriate to bring classical
and constructive mathematics into a competition according
to which of these disciplines reflect the mathematical
intuition in a better way. This would mean to adapt one
of the dogmatic standpoints of the originators.

This is a relatively fruitless approach and a more pragmatic
point of view is just to understand that classical and
constructive analysis reflect two different view points
(the non-uniform versus the uniform).

In computable analysis we treat both points of view in a
uniform model and we see this as an enrichment of views
and not as a competition.

Vasco Brattka
http://cca-net.de




> B) The failure in constructive mathematics of the intermediate
> value theorem *In its classical, intuitive form* makes
> the classical notion of a continuous function
> a better *approximation* of the original intuitive
> (or "naive", if one prefers. I do not mind) geometric
> concept. Note that this does not imply that the 
> constructive versions of the IVT 
> are not interesting or useful, or even intuitive from some 
> other perspective (I believe that they are).
> Note also that in my opinion the classical concept
> too fails to capture the original geometric notion!
> 
> Arnon Avron
>