FOM: for the record

[Randall Holmes]

I disagree with Friedman's statement

"ZFC is the complete formalization of mathematics"

or at least regard it as a "seriously misleading formulation".

The reasons that I object to this are similar to the reasons that he
objects to statements of mine along the lines of:

"first-order ZFC cannot express the notion "the set of natural numbers"";

my statement contains implicitly a notion of "express" which he
doesn't like, while his contains implicitly a notion of "mathematics"
which I don't like (and which I suspect would be unacceptable to many
readers of this list; as would my implicit definition of "express").

Given what I mean by the word "mathematics", his statement translates
to a falsehood, pure and simple.  However, on further examination of
the context, I can tell what he means (I still probably disagree, but
the disagreement is much less profound; it is in the area of sociology
rather than f.o.m.).

Mathematics is the study of abstract structures (my definition!), with
special attention to certain traditionally studied structures.
Con(ZFC) is an (admittedly rather bizarre) statement about one of the
traditionally studied abstract structures, so it is certainly a
mathematical statement.  By saying this, I am not disagreeing with
Friedman (though I probably do disagree, but that is another
story--see next paragraph).  But I could have challenged him on this
point without taking the trouble to figure out what he meant; please
take note!

He apparently means by "mathematics" the deductive closure of a set of
propositions accepted by the editors of mathematical journals for use
without special mention in journal articles.  I really don't think
this is a caricature; if Friedman does think so, I'd like some
expansion on his remarks in a recent posting to dispel this
impression.  In this sense, Con(ZFC) might not be a mathematical
statement.  But I'm not sure the Axiom of Choice is a mathematical
statement using this criterion (observing the behaviour of some
mathematicians).  Martin's Axiom is certainly not a mathematical
proposition by this criterion; from my standpoint it certainly is.

I suggest that in responding to one another's posts we take into
account what participants mean by commonly used terms.  I don't think
that it is reasonable to expect participants to abandon their
individual usages (there are no universally acceptable definitions of
such terms as "mathematics"!); but I think that we should both take
care to say something about our peculiar usages (as I did with
"express") and take care that when we dispute a proposition someone
has made we take the trouble to determine what they mean.

Friedman, by the way, gets points for this in the case of "express";
he did make it clear that he knew what I meant and criticized me on
that ground (which is OK).  But he does not get points for making an
extremely controversial statement which is based on an eccentric
definition of the word "mathematics" without making it clear that that
is what he is doing!

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes