FOM: understandability of Con(ZFC); bad PR
Stephen G Simpson
simpson at math.psu.edu
Wed Aug 26 20:44:22 EDT 1998
Joseph Shoenfield writes:
> the context of finite combinatorics. I replied that he was replacing
> f.c.p. by understandable f.c.p., and that the latter was not a suitable
> concept for our purposes, since the word understand is inherently vague.
I concede that "understand" isn't a mathematically rigorous concept.
But I don't agree that it's inherently vague. First of all, it's a
concept that we constantly apply in fairly precise ways, e.g. when we
sit in judgment of students. It's also crucial to philosophy,
psychology, pedagogy, and other respectable disciplines dealing with
the human mind from various perspectives. Perhaps it has more than
one meaning and these are in need of clarification; I think that's
what Neil and I have been doing. There are appropriate standards of
rigor in fields other than mathematics. (Let's talk another time
about whether f.o.m. is one of those fields.)
And yes, you are right that understanding is key to my argument about
Con(ZFC). My argument is that it's not an f.c.p. (= finite
combinatorial proposition) because human minds can't understand it as
such. You construe this as replacing "f.c.p." by "understandable
f.c.p." But I say that in its normal mathematical usage the term
"combinatorial" already has a component of understandability from a
combinatorial perspective. You can learn this by trying to convince
the editors of the Journal of Combinatorial Theory that statements
like Con(ZFC) are combinatorial and therefore would fit perfectly in
More generally, I would suggest that virtually any distinction between
two branches of mathematics involves understandability. For example,
we don't say that analysis is algebra, even though analysis can be
translated into algebra. I think the reason we don't say it is that
the translation doesn't preserve understandability. It seems to me
that if understandability is inherently vague, then so are all
distinctions between branches of mathematics.
Now that I think about it, I wonder if this could be a good
f.o.m. research project: to give a rigorous distinction between
algebra and analysis, not using understandability.
End of Digression ]
Anyway, putting the nomenclature dispute aside, don't you agree that
Harvey's statement is an f.c.p. (my term) or "understandable f.c.p."
(your term), while Con(ZFC) is not? And if this is the case, doesn't
Harvey's independence result represent serious progress in f.o.m.?
I guess you will dismiss these questions on the grounds that,
according to you, understandability is inherently vague and therefore
irrelevant. Very frustrating!
Let me try another tack. Do you at least agree that there is *some*
relevant qualitative difference between Con(ZFC) and Harvey's
> I regret to say that the reply of Steve ... was full of
> unsupported statements
It consisted mostly of questions, not statements (24 Aug 1998
> laced with emotional and prejudicial words and phrases.
Prejudicial words and phrases? Aren't you arguing over a matter of
mere terminology? Now, if you want to talk about emotional and
prejudicial words and phrases, how about "computability"? :-)
> I think the use of such phrases is due to the conviction that his
> opinions are true and important and his despair that he has
> difficulty convincing others whom he respects as logicians to agree
> to this.
Well, I'm not exactly in despair over this. But yes, you are right.
I do have a tremendous amount of respect for people like you and
Martin. And I do think my views are true, and I do wish I had more
success in convincing you.
> I only wish he could see that this sort of argument is the worst
> way to convince people of anything.
Ah well, yes, I know I'm not the greatest public relations guy. Thank
goodness I have Harvey to smooth things over! But Harvey, where are
you when I need you? :-)
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