FOM: the Borel universe/but it wasn't blather -- a problem about meaning
Robert S Tragesser
RTragesser at compuserve.com
Sun Dec 7 06:33:01 EST 1997
Yes, good from the point of view of thinking about CH. But the
blather was increasingly
only tangentially concerned with CH.
[1] The emergent very general foundational issue is:
When is a (proposed) mathematical problem (fully
and mathematically)
meaningful or definite?
Stuart Shapiro's quotation from Hilbert ( to the effect that a
question is definite only
if it is necessarily the case that it can be answered or shown to
be unanswerable) points
to a starting point for resolving that general foundational issue:
to give a rigorous
analysis and critique of Hilbert's criterion. N.B., it should be
an honest analysis and
critique -- by our best current lights -- and not what Hilbert
would have given. And it
should apply to all proposed mathematical problems and NOT require
their translation into
some canonical formal language (such as set theory). (The second
disjunct of the criterion
would certainly require something novel and deep.)
[2] The Borel example points to the necessity of distinguishing
between MEANING ( in something
like proof or truth conditions) and SIGNIFICANCE.
ZFC is supposed by some to be significant in that
its theorems tell us about pure sets of
which we are supposed to have some understanding
antecedently to ZFC (as achieved
for example through informally proven theorems -- theorems
proven more by the light of nature shining on
sets than about by appeal to explicit axioms).
Likewise, CH is supposed to be significant (beyond
its meaning qua problem about pure
sets) in that its proof or disproof is supposed to
give us a better understanding
of "intuitive" continua.
The last carries with it the assumption that the
representation of continua by pure sets
are in some sense true to intuitive continua.
As I think van Stigt on BROUWER and the relevant
essays (especially by WEYL)
in Mancosu's new anthology make clear, Brouwer found the
set theoretical analysis of
continua incorrect, as presumably revealed by
counter-intuitive (counter to intuitions
how continua and structures on them should behave)
pathologies.
Borel is clearly reactibng to the pathologies in
the same way; but he seems to think
that a sufficient representation of continua can be given
by restricting the classes of sets
entering into the reprentations. Whereas Brouwer required
a far more radical approach,
giving an alternative characterization of "SET". He
(apparently) settled CH (even trivially),
but under a radical transformation of its MEANING.
Borel (if I understand Steve Simpson's account)
wants to preserve the MEANING of
CH, but deploy his sense of its SIGNIFICANCE in order to
tighten down what counts
the set theoretic analysis/representation of continua.
This opens up the possibility of answering CH? when
its SIGNIFICANCE is
taken into account where as CH? in its (as it were) bare
meaning (as a quesition in pure
set theory) may be unanswerable.
Remark 1: Feferman's Principle (or Point of Wisdom).
Solomon Feferman's principle that in mathematics we get a
lot from a little is extremely important.
It can be used to suggest that thought that in general, in mathematics,
we should not expect, nor should we need, to mathematically explicate our
intuitive ideas (such as continuity or even set) fully. A little goes a
long way. The obverse side of this is that the state of our mathematical
tool-box at any particular time may limit the mathematication of an
intuitive idea. But later with an increase in our mathematical tools, we
may be able to capture quite other aspects of our intuition, and there is
likely nothing in the whole of being to force these two analyses to be
wholly compatible.
Remark 2: Brouwer's Anti-Realism. Whatever Brouwer's spiritual
commitments might have been, it is clear that his supposed anti-realism --
as far as his mathematics was concerned -- functioned to suggest an idea
which enableD him to give a mathematical representation of his intuition of
continua (qua fluidic continua ), viz. the idea of choice sequence, free
choice sequence (which were only partially mathematicized by Brouwer,
their full and successful mathematization arriviing via mathematical logic,
Kreisel/Troelstra et al). Choice sequence is then a tool thatg enable
Brouwer to given a mathematical representation of intuitive continua, of
their fluidic part. It might be worth observing that in a sense Charles
Peirce shared Brouwer's intuitive idea of continua in at least this respect
-- they are not resolvable into points. Pierce spoke of continua as being
"point sinks". From that point of view -- considering CH under its
SIGNIFICANCE -- the independence of CH and in partoicular the mata-theorem
that the power of the continuum can be almost any transfinite cardinal
EXPRESSES THE OBJECTIVE TRUTH OF THE MATTER!!!!!!
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