Practical Bivalence

[Harvey Friedman]

Periodically we have interesting discussions on the FOM concerning the
bivalence of mathematical propositions. Andthis is also a framework
commonly used in the general philosophy community. Of course, the
terminology "bivalence" is not used much within the mathematical
community but one can, with some effort, get some mathematicians
engaged in evaluating the bivalence of mathematical propositions.

However, I have never been comfortable with the framing of the
underlying issues - whatever they are - in these terms.

I would suggest that we see what happens if we replace bivalence by
what ai call Practical Bivalence.

Let P be a mathematical proposition P is Practically Bivalent if and only if

"it is a credible goal to attempt to prove or refute P in the same
sense that it is for the various standard well known open problems
throughout mathematics."

There may be better or clearer ways to formulate this kind of
Practical Bivalence.

We expect that for the foreseeable future there will be a steady dose
of unusually thematic and attractive combinatorial investigations
which have the following features. In low dimensions there are
beautifully satisfying proofs using unproblematic methods (e.g., well
within ZFC without power set), which are readily digested by a very
broad range of mathematicians working in a variety of areas. HOWEVER,
in higher dimensions, the statements are unprovable in ZFC and linked
to large cardinals. I.e., equivalent to the consistency of large
cardinals the strength of which increases as the dimension increases.

These developments could for some, leave the usual confidence in
Bivalence of arithmetic propositions in tact, but go against Practical
Bivalence of arithmetic propositions.

Harvey Friedman