[FOM] Classical logic and the mathematical practice

[Harvey Friedman]

On 5/10/05 3:43 PM, "Moshe David" <davidm2 at math.biu.ac.il> wrote:

> Dear FOM memebrs,
> 
> I'm a young researcher in complex analysis  and I have a bothersome
> question...
> If you could answer my question I would be grateful !!!
> 
> Q : Why the majority of the (working) mathematicians are still using
> classical logic ? to sharpen my question :
> we use the excluded middle without any worry and say that the real field is
> the disjoint union of  \Bbb Q and \Bbb Q^c
> though we know that there will be a chance that the rationality of some real
> numbers (e.g. the Euler constant) is undecidable assuming ZFC.
> Is the using of classical logic is not actually Realism/Platonism ? , is
> there any ontological or epistemological justification to use classical
> logic when we know
> that intuitionist logic is more safer and remote from Realism  ?
> 

Let me address just one small aspect of these questions. This concerns
"safer".

There are many results of the following form. Let T be one of the standard
formal systems based on classical logic. Let T' be the counterpart based on
intuitionistic logic. Then

T is consistent if and only if T' is consistent.

Furthermore, the proof of the previous line is carried out in a very weak
theory, far weaker than T and T'.

There are also a complex of results to the effect that

T and T' prove the same sentences of certain forms.

Mathematicians use classical logic in practice partly because they like to
avoid worrying about philosophical issues such as the ones you raise.
However, they have to think very explicitly when they try to give bounds or
algorithms. So they would rather stick with classical logic, and focus their
attention on explicitness or effectiveness within the framework of classical
logic.  

I should also add that mathematicians do get a sense of what is
"constructive" and have developed what they regard as an aesthetic in their
proofs. That is, generally they prefer to see a constructive argument than a
nonconstructive argument. They don't push this too hard, and do regard it as
more of an expositional matter than a substantive matter.

Harvey Friedman