FOM: Reply to Steve
Joseph Shoenfield
jrs at math.duke.edu
Tue Aug 18 16:23:47 EDT 1998
I am at a lossto reply to Steve's recent reply to me, since it seems
to me not to deal with the item in dispite. This dispute concerns the
statement: ConZFC is a statement of finite combinatorics. I think the
statement is true; Steve and Harvey do not. In his reply, Steve states
that first, "finite combinatorial statement", while not precisely
defined, is an informal notion which may prove useful in foundational
discussions; second, that use of such informal notions should adhere to
common mathematical usage; third, that "combinatorics can be described
informally but usefully as the study of the arrangements of objects in
patterns". I agree with all of this, especially the second point. I
would add a fourth point: ConZFC is a statement about the arrangement of
objects called sentences of the language of set theory into patterns
called proofs in ZFC. Steve has nothing to say about why ConZFC does not
fit his definition of a finite combinatorial statement. Harvey did have
one such remark which I understood to be that ConZFC is uninteresting
without motivation from quite different fields. This is what caused me
to say that he is replacing "finite combinatorial statement" by
"interesting finite combinatorial statement". I think my view here is in
accord with common mathematical usuage. For example, if one had a result
about the solutions of a PDE which was only interesting because of its
applications to biology, it would still be a PDE statement.
As to the dispute about "epochal", I am not convinced that we have
important disagreements about its meaning. I mentioned good logicians
because they could contribute to the epoch by proving interesting new
results using the concepts and techniques of the original epochal result.
Slightly modifying a statement of Steve, I think that a result can be
called epochal if it changes the way people interested in and knowledgable
about mathematics think about mathematics, especially if this revised
thinking led to a change in the way that they do mathematics. Our
difference is whether one should call a result epochal before one has
evidence that these two thing would happen which is convincing to people
who are doubtful. As I have previously noted, to do this will frequently
spur these doubtful people to search for evidence in the oppositie
direction.
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