FOM: on foundations etc.

[Lou van den Dries]

In response to Tennant's email:

"Whether mathematicians are still collectively exercised by what is to
count as a valid proof"?

Speaking only for myself of course: I don't think so. (There are of 
course people who strongly prefer proofs of a certain kind, like
"elementary proofs", because they feel they get more out of that.)
Anyway, proofs are judged mainly on other aspects  than correctness
(which is presupposed before we even speak of proof), like how much
informaton it gives, and how adaptable to other situations, how
"elegant", and economical, etc.

"Do they think of the meanings of mathematical terms as determined by the rules of inference (including those expressive of definitions governing them?---
or do they think of those meanings as having some other source, to which
the rules of inference are to be held responsible when judging of their
correctness?"

Hmm, these questions don't resonate with the way mathematicians think
about mathematics, at least it doesn't with me. How about considering
an example, say the term "group action". This involves a group, a set,
and an action of the group on the set. It has a perfectly simple
definition. What's important is that it captures to a large extent the
interaction of groups with other objects of mathematics, as 
permutation groups, groups of motions, automorphism groups, representations
of groups, etc. In other words, the notion "resonates" in the right way,
. That results about it have to be established respecting the rules
of inference goes without saying. (But while any system of rules of
inference will have something artificial about them, this is not the
case with the notion of group action, and the basic results on them.
Still, this doesn't mean the rules of inference are to be held 
repsonsible ... Rules of inference are so to say part of the 
"very-low-level" machinery that at this point of time is convenient in
mathematics, but real mathematics takes place on a higher level of
discourse. I am trying to use here an analogy with computer languages,
maybe incorrectly, but "rules of inference" is on the level of the
"machine language", and fortunately we rarely have to go back to that
level.)  Best regards,
                        Lou van den Dries