Matt Insall montez at fidnet.com
Sun Sep 5 13:12:59 EDT 2004

Timothy Chow wrote:
<<Yes, of course I understand that.  I guess it is always dangerous to be
even slightly informal with this audience!

Let me try a less offensive phrasing.  In a world where AC is true, we
expect to observe phenomena that violate our intuition about measure.
Freiling's argument shows that if CH is true, then we observe a phenomenon
that violates our intuition about probability.  So what?>>

It seems to me that in this crowd it is also dangerous to not be informal
enough.  :-}

I have not read Freiling's work, so some of the following is just a hunch,
based upon discussions I have had in the past and things I have read, about
what science is about (as opposed to mathematics and FOM).

The question is really one of which intuition should rule:  the one proposed
by Freiling, or an opposing intuition.  The original posting mentioned
Freiling's ``evidence'' against CH.  I guess that now that I have thought
about it, I would not consider it ``evidence'', but ``support'' of one
intuition as opposed to another.  Scientifically, one chooses theories that
not only explain what seems to want explaining, but explain it in a way that
is as intuitive and simple as possible.  (I am probably setting myself up
for a lashing here, because I am being ``informal''.  Sorry, I have not
studied the entire lexicon of the philosophy of science.)  The language of
probability is used extensively in the sciences, and intuition about
probability is of paramount importance in providing useful theories about
nature and explanations of natural phenomena.  Thus, in the presentation of
an argument (not a formal proof) in favour of an hypothesis, one invokes
_preferred_ intuitions.  Apparently, Freiling has a reason for _preferring_
the intuitive ideas behind the stated probabilistic argument against CH.
The question was why do set theorists not like this ``evidence'', and I
think by extension, the question really leads to ``why do set theorists not
like this _preference_'', which is based upon the scientific preference of a
simpler, more intuitive theory of physical phenomena.  Even though AC on the
one hand violates our intuition about measure, it also on the other hand
satisfies our intuition about some other things, and we have made a choice
about which intuitive concepts we want our set theory to support (namely
those in which AC holds).  It seems to me that set theorists have not made
such a choice relative to CH, so the question of why they have either denied
or ignored the intuitive content of Freiling's argument against CH is a
natural one:  It comes down to a question of what is the basis for choosing
one axiom or another in mathematics (or, more specifically, in set theory) -
intuitions that provide mathematics that can be used to develop scientific
explanations that are simpler and more intuitive, or ...?

Dr. Matt Insall
Associate Professor of Mathematics
Department of Mathematics and Statistics
University of Missouri - Rolla
Rolla MO 65409-0020

insall at umr.edu

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