[FOM] Classical logic and the mathematical practice

[Arnon Avron]

On Wed, May 11, 2005 at 10:26:16AM +0200, Jeremy Clark wrote:
> 
> I think that the majority of working mathematicians are still using 
> classical logic because there is a tradition of using it which is 
> difficult to escape from. I cannot see any other reason for favouring 
> classical over constructive mathematics. 

Can't you really?

So let me give you two reasons.

One simple reason is that most mathematicians are platonists. They
believe that what they talk about is meaningful, and that propositions
about it are either true or false. And for people with such views 
classical logic is simply *valid*, beyond any doubt.

The second (more important reason for me personally, because I am not
a platonist concerning the "real numbers"), is that classical analysis 
is by far more faithful to the geometrical intuitions on which analysis
is based. These intuitions are indispensable for most mathematicians
(and maybe for all physicists), and in my opinion provide the only
possible justification for using any analysis (either classical or
"constructive"). For example: our intuitive notion of a continuous
function is that it is a function the graph of which can be drawn
without leaving the pen from the paper. For this intuitive notion
it is absolutely obvious that if f(a)<0 and f(b)>0, then there is
a<x<b such that f(x)=0. This is indeed a theorem of classical analysis
concerning its official notion of a continuous functions, but it is
not a theorem (and there are "counterexamples" to it) in the various
constructive systems of analysis. Therefore (and because of other
theorems that are geometrically obvious) the classical notion of 
continuous function makes sense and is acceptable to the majority
of mathematicians, while the constructive one is not.
 To summarize: analysis was originally based on, and is still guided 
by, geometrical intuitions. "Constructive analysis" completely ignores
these intuitions, and so it is  little wonder that most mathematicians
ignore it in return.

Arnon Avron
School of CS
Tel-Aviv University