FOM: finite combinatorics; what is "epochal"?
Stephen G Simpson
simpson at math.psu.edu
Mon Aug 17 15:21:22 EDT 1998
Joseph Shoenfield writes:
> I think Harvey is replacing the notion of finite combinatorial
> statement by the notion of an interesting finite combinatorial
I don't think that was Harvey's point or intention. I think Harvey is
simply relying on normal mathematical usage. The distinction to be
drawn is not between "finite combinatorial statement" and "interesting
finite combinatorial statement", but rather between "non-combinatorial
statement" and "combinatorial statement".
We need to get clearer on these distinctions.
Shoenfield asserts that Con(ZFC) is a finite combinatorial statement.
I understand the sense in which this assertion is meant, but we need
to recognize that this assertion is significantly at odds with normal
mathematical usage. According to normal mathematical usage,
combinatorics is a certain branch of mathematics, sometimes described
informally but usefully as being concerned with arrangments of objects
in patterns. Other branches of mathematics are: algebra, geometry,
mathematical logic, etc. The boundary lines among these branches are
informally drawn but reasonably sharp and scientifically valid. All
of this is part of normal mathematical usage, understood by virtually
every mathematician. By contrast, when Shoenfield says "finite
combinatorial", he is evidently referring to what logicians might call
a formal explication. Probably what he has in mind is Pi^0_infinity.
This formal explication of "finite combinatorial" is in some ways
useful, but it has its limitations; in particular, it does not respect
distinctions among branches of mathematics as above, e.g. the
distinction between combinatorics and mathematical logic. Shoenfield
is choosing to ignore these informal distinctions, which in my view
are essential to the current discussion.
Despite the importance of rigor and formal definitions in mathematics,
it seems to me that there are also a lot of informal notions that are
also extremely important and relevant, especially when it comes to
evaluation of research in f.o.m. and other mathematical subjects.
Among these are: "simplicity", "elegance", "concept", "geometrical",
"algebraic", various branches of mathematics, etc etc. Perhaps we
need to discuss these informal notions and their role.
> I don't think the opinions of finite combinatorialists is relevant
> to the foundational significance of Harvey's result, since they are
> not knowledgable about fom.
Yes, I agree to a certain extent. When I mentioned finite
combinatorialists, my purpose was only to focus attention on the
useful but informal distinction drawn above, between combinatorics and
other branches of mathematics. So long as we can recognize and
understand this distinction, there is no particular need to consult
finite combinatorialists about the interest of advances in f.o.m. My
question is, does Shoenfield recognize and acknowledge the relevance
of this distinction?
> Of course, it might happen that consideration of the notion of an
> interesting finite combinatorial statement would lead to a precisely
> defined notion which differentiates the tree principle from ConZFC and
> which is foundationally signficant. ...
Yes. However, in the meantime, such a precisely defined or formal
notion is not needed in order to appreciate the importance of
Harvey-style independence results. The informal distinction between
"combinatorial" and "non-combinatorial" is sufficient. Can we agree
on this informal distinction?
> I would have objected much less to Steve's original statement if he
> had used "important" instead of "epochal", or at least showed why
> he considered epochal more appropriate. To me, epochal suggests
> the beginning of a new epoch, in which many good logicians would
> utilize the concepts and methods of the epochal result to prove
> important mathematical results.
This makes me think that Joe Shoenfield and I disagree about the
meaning of "epochal". For Shoenfield, "epochal" is measured in terms
of impact on "good logicians", and a new epoch is an era in which
certain new concepts and methods are utilized to prove important
mathematical results. To me, "epochal" refers to impact on the entire
intellectual enterprise, not limited to mathematical logic. I
described Harvey's independence result as epochal, because it may
change the way people think about mathematics. If the incompleteness
phenomenon is extended into the realm of finite combinatorics and
other realms of mathematics which were hitherto immune, then that may
represent a new era in mathematical thought.
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