FOM: generic absoluteness
Mark Steiner
marksa at vms.huji.ac.il
Wed Jan 26 11:11:03 EST 2000
The posting by John Steel really struck a chord with me. First of
all, his account of the dispute among specialists between those who want
to
restrict mathematical hypotheses to consistency and those who want to
assert their truth, reminded me of similar debates among "scientific
realists" and "empiricists" in the philosophy of science. For example,
Bas Van Fraassen (Princeton) argues that it is always more rational to
accept a theory in science as "empirically adequate" (i.e. consistent
with human observations), which is a weaker hypothesis than that it is
true.
Second, his powerful summary of the case for the naturalness of
PD
and the fact that "all roads lead to PD" recall similar episodes in this
history of mathematics when various expansions of mathematical concepts
seemed forced upon us (and I have already mentioned, in previous
postings, a new ms. by my colleague Meir Buzaglo, on the concept of
"forced expansions" in mathematics).
The concept of "reflection" of the expanded concept down into
the
more restricted one which he mentions also reminds me (perhaps by free
association) of cases where it looked like mathematical theories
contained hints concerning their own expansion. (I'm thinking of cases
in complex analysis, where mathematicians spoke of "multiple valued
functions", even though they knew there strictly speaking is no such
thing, because they believed that in a suitable expansion (such as the
later Riemann space) the function would find its "true home" and be a
natural single valued function. I like to say that mathematical
theories contain "latent information" about expansions.
Usually, when a concept has been both "natural" and "forced upon
us," however, it also turns out to be connected with many other branches
of
"core mathematics." So we should "expect" that PD should be usable to
decide open problems in "core" mathematics. (Then, later, we may or may
not be able to prove the same thing without PD.) I have the feeling,
however, that "core" mathematicians are not running to learn set
theory. Is it because, as Harvey thinks, that we have simply not
developed "core mathematics" enough? But even if every present "core"
problem is decidable in ZFC, it might well be possible that, like the
case of Borel determinacy, there could be many mathematical conjectures
whose proof does not need PD. And unlike Borel determinacy (thanks
Steve for the clarification of this point), there might be theorems of
core mathematics whose proof is most natural, simple,or even explanatory
using PD. Where are they?
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