FOM: Internal and External Characterizations of Mathematics

[Shipman, Joe x2845]

   I offered suggestions to amend Hersh's "definition" of mathematics because it
would be very interesting to have such an "external" characterization.  I
believe that the essence of mathematical thought is somehow related to its
internal structure, so I would then still decline to call a Hershian
characterization a "definition", but if the characterization covers all and
only what we call "mathematics" then we have to explain this coincidence so
that we can rebut claims about what we are "really" doing.  (In his turn, Hersh
is faced, as he admits, with the problem of explaining why an area of human
thought meeting his charaterization exists.)
   Machover is of course correct that my chess statement can be formalized
mathematically and we think we know what would qualify as mathematical proof of
it.  But Hersh doesn't want to get rid of proofs like Zermelo's when he says he
should exclude chess, he just wants to say that the way chessplayers know my
statement is true is not mathematical!  For this he needs to distinguish proofs
from investigations.  Machover's insistence on the unique certitude of properly
derived mathematical results is not enough, because you can't distinguish the 
mathematicians from the chessplayers by the degree of certitude (I personally
would be less surprised to see an inconsistency discovered in ZF than to learn
that White could not force a win when Black's Queen was removed).
  What I have been trying to elicit here is an "internal" characterization 
of mathematics (which I might hope to recognize as a definition) to answer 
Hersh's social constructivist claims with.  The mathematicans on this list
may feel they know mathematics when they see it (or do it) but in the 
absence of a good definition of what mathematics is we are vulnerable 
to philosophers telling us what it is we are "really" doing (this is 
not meant to suggest that Hersh himself is not a mathematician)!
-- Joe Shipman