[FOM] Thoughts on CH 3

[Harvey Friedman]

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THOUGHTS ON CH 3

Sol claims in his paper that CH is "inherently vague". He first uses this
phrase on page 19 of his paper. (Sol's call for comments on this paper
started this discussion group).

I said that I would agree to the extent that CH research is not a
relatively promising area of research in the foundations of mathematics. Of
course, CH research is, and will continue to be, of interest to some or
many experts in the set theory community. And that, of course, will be
enough for the set theory community to sustain it - as long as there
remains a professional set theory community, which is doubtful in the long
run on the basis of what is going on now. Historically, math sheds areas
that do not substantially interact with other areas. That perhaps would not
be the case if math didn't continually produce striking interactions
between areas. But math is very strong in this respect, and there is no
prospect for math giving up on the deep interaction requirement.

However, the prospects for interest in CH research to jump to either the
math or the philosophy or any other community are not (relatively) good. It
is relatively easy to get some recognition in wider communities for a
limited time until the drawbacks and limitations eventually become well
known, and well accepted, and people move onto other things.

Rather than go on in detail about drawbacks/limitations here - because I
want to get back to Sol's "inherently vague" - let me just say that the
overall difficulty with CH research is its complexity - not the proofs,
which is NOT the problem, except that it might cause some practical
logistical issues - but the complexity of formulations. In order to get and
sustain the genuine interest of scholars who are not experts, there has to
be simple and honest stories. Much simpler than what experts who have
immersed themselves in for very long periods of time, and much more honest
than sound bites. Much simpler even than dedicated Ph.D. track students in
top intense programs. The simpler and more honest the stories, the greater
chance for sustained general interest.

In some of this research, countable models of set theory are used in
formulations, and unless alternative formulations are emphasized, this is
ultimately a non starter for a philosopher looking for "philosophically
fundamental" or "philosophically compelling" developments. "Mental
pictures" and "maximality" are familiar old (and good) ideas, at least in
universally fundamental contexts. However, it is not yet clear how
appropriate these concepts are when they are awkwardly and uncritically
forced into technical contexts. I will say much more about this if there
are responses about this. In fact, I am nowadays trying to do some "mental
picture" research at a fundamental level, and also try my hand at
"maximality" research at a fundamental level.

It may be possible to make some of the CH research more philosophically or
foundationally compelling, and I stand willing to take a serious look at
this, cooperatively. However, thus far, I do not see any interest in the
relevant parties presenting and frankly discussing the philosophical and
foundational weak points in order to get new philosophical and foundational
ideas. This is backward, in that there are dozens of experts on this list
in philosophy, foundations, and math logic, some of which are absolute
world class icons. In fact, the failure to use this list to clearly state
the essence of the research, including weak points, in clear terms, looking
for new ideas among people on the list, can be viewed, ultimately, as
suspicious.

Back to Sol's "inherently vague". I said that "inherently vague" is or
seems to be at this point, "inherently vague". Sol responded by saying that
in the case of CH, his reasons for it being "inherently vague" are not
"inherently vague". But that does not mean that the general concept
"inherently vague" isn't "inherently vague" - unless you accept that Sol's
account of why CH is "inherently vague" is generally applicable.

But I think that (someone like) Sol could equally well adjust his framework
and discussion to argue that "the ring of all integers is inherently
vague". Of course, the criteria here would have to be much stricter - much
stricter in order for something to "have a definite truth value":.

I think that for most people, the incompleteness phenomena plays a critical
role in the background - at least in the practical sense. Consider the
following two assertions:

1. CH is inherently vague.
2. "There are no odd perfect numbers" is inherently vague.

I have heard some people, including Sol, subscribe to 1. However, there are
some people who are not aware, or who have forgotten, that CH was shown to
be independent of ZFC. And one can survey people who are old enough to
remember what things were like before Cohen. I think that nobody unfamiliar
with the independence of CH from ZFC would even think of asserting 1. Yes,
after the fact, one can essentially diagnose the independence of CH, and
look for reasons to doubt the "legitimacy" of CH.

However, I don't know of anybody who asserts 2 - except some finitists and
ultrafinitists. Actually, I think that you can argue that many if not most
people on the planet agree with 2 in a sense - as they are not
mathematicians. I think that most people on the planet that you could get
interested enough to have a conversation, would find the following more
persuasive than other more common points of view among mathematicians.

"There is no right or wrong about 2 because not all numbers exist at the
moment - and this will always be the case in the future. When we find and
verify an actual odd perfect number, or when we prove there are none by an
obviously correct proof using universally accepted reasoning of the
clearest kind, then the statement becomes true or false, and only then.
Until then, the statement is only a research project."

or some variant of such. I would like to know if this is the most
attractive position for a) theoretical computer scientists b) applied
computer scientists c) theoretical physicists d) applied physicists e) news
commentators f) congressman. I don't think it is the most attractive
position among mathematicians and philosophers, but I also wonder what the
numbers are.

Now suppose that there were say a dozen results that various well known
arithmetic problems like 2 were equivalent to the consistency of various
large cardinal hypotheses. Then I submit that some mathematicians and
philosophers would feel compelled to talk about "inherent vagueness" in 2.
This is especially clear if there was some general shape of an arithmetic
statement that somehow became indicative of its being equivalent to the
consistency of large cardinals. In the case of set theory, this general
shape is already apparent to logicians and intuitively apparent to
mathematicians - high levels of generality, involving objects of an
unusually non concrete nature.

Harvey Friedman
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