FOM: Mathematics, logic and the rational

[Vladimir Sazonov]

Doug McKay wrote:
> 
> A few recent comments seem to suggest that many contemporary mathematicians
> don't feel the need to bow to logic. How can that be? Isn't mathematics the
> principle example of rationality? Or has it become a "here a miracle
> happens" sort of endevour? How can you sidestep logic and call yourself a
> mathematician? The mind boggles.
> 
> If mathematics isn't a "rationally coherent structure" then what is it?

No doubts, mathematics is a "rationally coherent structure" and is
_based_
on (in general _any_ kind of) logical reasoning. Every mathematician
uses
modus ponens, reductio ad absurdum and other logical rules without even
thinking about this very much. This is like the atmosphere without which
the life is impossible. But who cares when there is no lack of it? There
was a critical situation e.g. with continuum hypothesis. Now, almost
every
mathematician knows that it is mathematical logic which was used to show
its independence. This mere knowledge is usually quite enough without
any
real need to go into details on forcing etc.

The attitude of the "ordinary" mathematicians to mathematical logic may
be changed when more and more notions/problems/proofs important to them
(and to other scientists) will be crucially based on the corresponding
explicitly presented logical techniques. (Pure results, even such
as independence of CH are, of course, not enough for this.) By the way,
if the subject of f.o.m. is just foundations of the (successfully!)
existing mathematics then the discussed negative attitude will be surely
preserved. Either we should wait when new parts of mathematics will
arise which will need a help of logicians, or we would try to foresee
these critical directions (where e.g. "new foundations" different from
ZFC will be urged).

On the other hand, mathematical logic as a part of mathematics may
co-exist very well in some (actually partial) isolation from the rest
of math. because it also has close relations with the external world
(just computers) and has its own internal life.


Vladimir Sazonov
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