[FOM] My personal experience with LEM

[Nik Weaver]

Martin Davis wrote:

> Noting that this property of not being closed under taking complements
> was shared by the two classes of r.e. and of Diophantine sets, I was
> encouraged by this blatantly non-constructive proof to conjecture that
> the two classes were one and the same. A proof of this conjecture, and
> hence a constructive proof of my theorem as an immediate corollary, had
> to wait 20 years for Yuri Matiyasevich to produce the crucial final
> step culminating an effort by Hilary Putnam, Julia Robinson, and
> myself.

I take the point of this message to be that although a constructive
proof was eventually found, it took much longer, was far more
difficult, and was achieved along lines suggested by the original
nonconstructive result.

It is possible to take the position that evidence of this kind does
not bear on the question of whether LEM is actually correct.  My
opinion (just an opinion) is that this may be true in principle,
but because philosophical reasoning is so insecure --- we lack
either a rigorous standard of proof or a way of testing hypotheses
experimentally --- it behooves us to take indirect evidence like
this seriously.

I wonder if a good argument could be made against my conceptualist
views in a similar way.  For instance, are there any good examples
of important mainstream theorems whose proofs would have been
substantially retarded if uncountable/nonseparable objects had not
been available?

The evidence from my specialty, C*-algebras, is rather in the
opposite direction: the nonseparable case has, generally speaking,
been an irrelevant distraction.  I think most students in functional
analysis initially think that nonseparable spaces are much more
important than they actually are --- there is nothing in the
definition of a Banach space (or topological vector space, or
C*-algebra, etc.) which would tell you that the nonseparable case
has little value.  This is something that people only learn with
experience.

My personal experience includes the solution of two long-standing
open problems in the "wrong" direction because of nonseparable
pathologies.  Dixmier's problem asked whether every prime C*-algebra
is primitive, and there turns out to be a pathological nonseparable
counterexample (N. Weaver, J. Funct. 203 (2003), 356-361); Naimark's
problem asked whether the only C*-algebras with only one irreducible
representation are the algebras of all compact operators on a Hilbert
space --- here the result is that the existence of a nonseparable
counterexample is consistent with ZFC (C. Akemann and N. Weaver,
Proc. Nat. Acad. Sci. USA 101 (2004), 7522-7525).  In both cases
a positive result in the separable case had been known for decades,
and there had apparently been a substantial amount of wasted effort
expended trying to prove the same thing for nonseparable algebras.
These examples are hardly in the same category as Davis's solution
of Hilbert's tenth problem, but I do think they are significant.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu