FOM: constructivist philosophy

[Fred Richman]

Stephen G Simpson wrote:
 
> A response to Fred Richman's posting of today.
> 
>> It seems to me that the philosophical discussion of intuitionistic
>> systems should focus on how they are currently viewed and used, not
>> on some "original intent".
> 
> Why not focus on both?

I suppose it has to do with the meaning of the word "focus".

> For example, we know that Bishop has stated (rather vehemently) that
> nonconstructive existence proofs constitute fraud.  Why?  If we prove
> the absurdity of (forall x) not Phi(x), why does Bishop think this is
> not a proof (or at least very strong evidence) that (exists x) Phi(x)?

I imagine that Bishop would have taken such a proof as evidence even
if he though it constituted a fraud. He did say that every classical
proof was a challenge to constructive mathematicians. Just like every
inelegant proof, or overly specialized theorem, is a challenge to
classical mathematicians.

I'm not sure there are any really good examples of this kind of proof.
One might think of the intermediate value theorem, where Phi(x) is
f(x) = 0. There is a constructive proof of the absurdity of "f(x) is
different from 0 for all x", but "f(x) is different from 0" is not the
simple negation of f(x) = 0. (Whether there is a constructive proof of
the absurdity of the weaker statement seems terribly uninteresting to
me.) For decidable Phi, this is Markov's principle, which is accepted
by some constructivists. Apparently, every time you have a
constructive proof of the hypothesis, you can get a constructive proof
of the conclusion. Is this reason enough to adopt the principle? That
seems like a philosophical question that doesn't involve "a
comprehensive integrated world-view of man, nature, ... etc.". A
somewhat similar situation is Church's thesis in a constructive
setting. Nobody has come up with a function that is not recursive, nor
does it seem likely that anyone ever will. Is that reason enough to
postulate Church's thesis?

I think the Brouwerian counterexamples to the intermediate value
theorem are the arguments against classical proof. Of course they do
not speak to everybody.

> The moon exists, yet we did not construct it, and no fraud is
> involved.  Why and how does Bishop think mathematics is different
> from the rest of science?

I have always been a little mystified by the inclusion of mathematics
with the sciences. It seems bizarre to me to draw a parallel between
the existence of the moon and the existence of an odd perfect number,
while the question of the existence of the number 5 strikes me as
almost meaningless.

> I don't see these issues (hard vs soft analysis, objects vs maps, etc)
> as truly philosophical.  They do not seem to flow from any
> comprehensive world-view or conflict of world-views. ...
> As they stand, I would identify them as relatively narrow,
> methodological issues.

You could say that hard analysis sees mathematics as problems to solve
while soft analysis sees it as a situation to understand. This is not
just a methodological issue---they are striving to accomplish
different things and their criteria for success are different.
Category theory, like constructivism, is a way of looking at all of
mathematics. In both cases, when you adopt the point of view, some
mathematical questions become uninteresting and others become very
interesting. Such comprehensive views of mathematics seem truly
philosophical even if they cannot be extended to a comprehensive
world-view (which the hard vs soft analysis probably could be).

--Fred