FOM: Friedman on Realism/Philosophy (reply)
steel@math.berkeley.edu
steel at math.berkeley.edu
Fri Jan 30 13:14:31 EST 1998
This is a belated partial reply to a post of Harvey Friedman from
1/18/98.
I had written earlier (1/15/98):
As a philosophical framework, Realism is right but not all that
interesting. Both proponents and opponents sometimes try to present it as
something more intriguing than it is, say by speaking of an "objective
world of sets". Such rhetoric adds more heat than light.
To which Harvey replied:
I think that you underestimate how intriguing it is. After all, you claim
to be working on the continuum hypothesis. Certainly nobody other than a
small minority of professional set theorists are working on it at all. And
you even have confidence that you are not wasting your time investigating
it from a Godelian naive realist point of view. Wow! And you are a naive
realist all the way through the cumulative hierarchy, right? So, it is not
a matter of choice of humans how far to go up the cumulative hierarchy.
There is merely "all the way" and any proper initial segment that one may
for special reasons restrict attention to, right? I.e., there definitely
is a measurable cardinal or there definitely is not a measurable cardinal;
and in fact, we know that there is a measurable cardinal, right? I think
that you're beliefs are very intriguing.
Here are a few comments:
1. It is "there are sets" which I claimed is not very intriguing. "There
are sets" is, by itself, a pretty weak assertion! Realism asserts that
there are sets, and hence (practicing Quine's "semantic ascent"), that
"there are sets" is true. If we want to puff up this last assertion so as
to make it seem more profound or have more structure, I suppose we could
say "The assertion expressed by "there are sets" is part of our best
theory of the world". I don't see any difference between these assertions.
Whether this is Godelian naive realism I don't know.
2. I don't claim to be working on the Continuum Problem.
3. I and many others do think "there are measurable cardinals" is an
intriguing assertion. That's why the possibility has been investigated
so thoroughly in the last 40 years!
4. I would subscribe to "either there is a measurable cardinal or there is
no measurable cardinal". (I don't see what the word "definitely" adds
to this assertion.) Harvey, do you accept all instances of the law of
the excluded middle? If you were able to prove some sentence phi of
the language of set theory from principles you accept by taking cases
on whether or not there is a measurable cardinal, would you regard
phi as having been proved?
5. I would say that there is good evidence that there are measurable
cardinals. "Know" is too all-or-nothing.
On approaches to deciding the CH, I had written earlier:
One man's technical jargon is another's deep insight.Immediacy is not
necessarily the same thing as being fundamental. We are probably past the
days when new axioms for set theory will be as easily understandable to
the layman as the Zermelo axioms.
To which Harvey replied:
I think you underestimate just how bad this is for your point of view. You
may have to eventually compete with things being done in a variety of
subjects inside and outside f.o.m., and inside and outside mathematics,
which are understandable to the layman compared to what you have in mind,
and you may suffer greatly in the inevitable comparisons. It's a serious
problem for your approach.
This seems to me just public relations. We should follow the terrain
where it takes us. How easily understood are the basic principles of
Physics? Should physicists add to their requirements on a "theory of
everything" the demand that it be easily understood by the layman?
Concerning neo-relativism, I wrote
>2. One ambition in foundations is to construct a universal framework
>theory in which all mathematical theories can be naturally interpreted.
>We want to have ONE picture, so that, as Harvey put it, "people can
>work together on a common ground".
To which Harvey replied
In pictorial set theory, it is hopefully going to be a theorem that all
pictures are compatible in some interesting sense. E.g., if they contain a
base part about N, then the sentences that follow about N in the pictures
are always compatible under inclusion.
The optimistic view is that we can put all pictures together
into ONE picture in some useful way. It seems to me that it is Harvey who
is "willing to settle for less" at this point.
Harvey continues with a description of "pictorial set theory" with
which I need some help. So here are some questions:
1. Should we think of a picture as an interpretation (in the informal
sense of meaning-assignment) of the language of set theory?
2. Is the set of sentences true in a given picture sometimes/always/never
an r.e. set?
3. Does every picture have all instances of the law of the excluded middle
in the language of set theory?
4. Is it anyone's job to relate different pictures to each other, and in
particular to put them into a common framework?
A bit earier Harvey writes
I don't think there is any "picture" that "solves" CH, and that should be
a theorem. However, you may think so. And what I worry about is that there
is a picture of yours that proves that c is real valued measurable, and a
picture of yours that proves MA. They are incompatible. In fact, I can't
tell whether or not, from your point of view, which of these two
statements is true and which are false. Or how you intend to decide them.
One hope is the following. Generic absoluteness provides a test of
"practical completeness". For example, it is plausible that all statements
about L(R) can be decided by large cardinal axioms, and one strong
evidence of this is the fact that if there are arbitrarily large Woodin
cardinals in V, then L(R)^V is elementarily equivalent to L(R)^V[G] for
all G set-generic over V. In other words, there are no independence
results provable via forcing concerning the theory of L(R), granted
arbitrarily large Woodin cardinals. ( Of course, there are Godel
incompleteness independence results, but large cardinals remove these.)
This kind of generic absoluteness is impossible for the theory of
L(P(R)) by Levy-Solovay.(CH is a first order sentence about this
structure.) But one can hope for an axiom/hypothesis A such that
1) A is compatible with all large cardinal axioms
2) granted sufficiently large cardinals in V, any two set-generic
extensions of V satisfying A have elementarily equivalent L(P(R))'s. I
think that finding such an A which is compatible with axioms
working similarly for L(PP(R))), etc., may amount to solving the
Continuum Problem.
There may be another axiom B with all the properties of A, but
incompatible--maybe even deciding CH differently. But A and B should be
provably consistent relative to some large cardinal axiom by forcing. In
the most natural case, there would be a definable, homogeneous partial
order used. In this case, the A-believer can think of B as the theory of a
certain sort of generic extension of his universe, and the B-believer can
think of A similarly. The theories are "generically bi-interpretable". In
the imagery above, we know exactly how to put the theories together.
Developing one is the same as developing the other. So I think it is too
strong to say that we would have incompatible pictures at that point.
John Steel
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