FOM: Reply to Wallen on "proof"

[Lincoln Wallen]

   Date: Tue, 10 Mar 1998 13:43:41 -0500
   From: jshipman at bloomberg.net (JOE SHIPMAN, BLOOMBERG/  SKILLMAN)

   This does need further explication on FOM.  The reason everyone (well, many
   people anyway) took issue with Hersh was that according to his sociological
   conditions proof was nothing more than the process by which mathematicians
   arrive at a reproducible consensus about abstract statements.  Hersh was
   misunderstood as failing to recognize the uniquely objective character of
   mathematical arguments as opposed to other types, but in fact he not only
   recognized this, he used it as his definition of mathematics. What he declined
   to do is EXPLAIN what was special about mathematical proof despite proddings by
   Machover and myself.  I used an example from chess to show that his purely
   external definition of mathematics was insufficient to characterize it; even if
   there were a better external definition (I suggested one based on the nature of
   communicability of proofs) we still need a characterization of mathematical
   proof that is based on what a proof is, internally, rather than the
   sociological effect it has of reliably and reproducibly inducing consensus
   about the statement to be proven.  -- Joe Shipman

I think Hersh was somewhat misinterpreted as propounding a social
constructivist view of mathematics which would, as you say above, have
us believe that "mathematical" consensus is the defining character of
mathematics (and any other professional human activity).  I agree that
one thing which is needed is an explanation of what is special about
mathematical proof.

I am less sure that I agree that an "internal" characterization of
math. proof, which in the above I understand you to mean something
which is *not* based on the fact that math. proofs have an effect of
"reliably and reproducibly inducing consensus about the statement to
be proven", will be foundationally adequate.  Unless by "foundation",
we agree to mean an explication to be formulated within mathematics
itself (f.o.m. as a branch of mathematics, at least as far as the
notion of proof goes), and I would not like to take this step at the
moment (unless you have a compelling argument for taking it).

Might not asking the question; "what sort of consensus?" is induced by
mathematical proofs be fruitful?  For example, one inadequate answer
to this sort of question might be "the in principle formalizability of
a mathematical proof."  The question remains of how this property is
achieved by a combination of the form and structure of math. proofs
taken as texts to be read (perhaps close to your notion of "internal")
together with the structure and properties of the practices for
reading such texts: for it is certain that math. proofs are written to
be read in distinctive ways so as to make their point.  (A possibly
related point: I remember Kreisel being concerned somewhere that
math. proofs not be massaged before attempts are made to unwind
them---I forget the reference, I understood his concern to be that the
results of unwindings should tell us something about the math. proof
as produced, not some other "version" of it.)

I am not advocating the above, highly naive, answer (formalizability),
just putting it forward as a subject for discussion.  I am aware of
some of the attempts to advance and refute it.

I will stop here to see your response as you may disagree with some or
all of the above, but it should be clear that although I can agree
that Wittgenstein provided few adequate answers to questions of this
kind, at least he raised some of them.  I am interested in hearing
what progress is being made addressing some of these issues, and
evaluations of that progress.  I also appreciated Hersh's views as a
means of again at least pointing to unresolved issues, despite the
substantial developments within *mathematical* logic.

Now to be perhaps provocative --- so please ignore this if it would
detract from your commenting on the above --- the recent discussion
over set-theoretical vs categorical foundations would seem to me to be
unresolvable in advance of addressing the question about the nature of
math. proof. (and of course use of language).  Consequently the
protagonists have resolved these questions in some way; understanding
this resolution (or resolutions) may, I only say "may", illuminate why
there is such disagreement, and allow those of us who are not
currently able to take a position (am I the only one?) with any degree
of conviction to advance our understanding.

Lincoln Wallen