FOM: Internal and External Characterizations of Mathematics

[Charles Silver]

Joe Shipman wants to investigate Hersh's "definition" of mathematics both
internally and externally:

Joe Shipman:
>    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)! 

	In a very general sense, I don't see that Hersh is in such deep
water.  In terms of a characterization I favored earlier, it seems to me
there are two sides to his view about mathematics.  The two sides are: 
Agreement and Aboutness.  "Agreement" is all about social consensus; 
"Aboutness" is what mathematics is about.  I think they correspond to your
internal/external distinction.  I think Hersh would need to add a number
of distinctions that I don't believe he presently makes in order to clear
up the relation between the two kinds of things (int/ext or Agr/Abo). But,
it seems to me he could work both sides of the street on this (though this
*could* violate some of his basic principles, I'm not sure).  For example,
he could work the Aboutness (internal) side by saying lots of things that
characterize mathematics in terms of the accepted subject matter of math.
No problem there, I think.  Then, working the external or Agreement side,
he could also say lots of things about the nature of mathematical
agreement.  He could do this rather sociologically, which I think is the
spirit of his analysis in his book (which I have not been able to get so
far). The novelty in his approach, I think, is that he thinks the nature
of mathematical consensus is a major factor in determining what
mathematics *is*.  Since what mathematics *is* would already have been
explained internally--according to this suggestion--he'd have to establish
a kind of bridge between the two, a kind of "if and only if" between the
two sides that would forge them together.  But, maybe if he looks
carefully at a number of issues that you and a number of others have
raised, he'll want to weaken the relation to just "if".  And, he may also
wish to add various qualifications.  I suggested looking at a "causal"
link between the two sides, though I happen not to like causal
explanations.  At any rate, it seems to me <*in general*> that he is not
in such hot water, though I think his analysis would have to become a lot
more sophisticated. 

Charlie Silver
Smith College