FOM: meaning, significance, CH, Tragesser; some positive remarks

[Stephen G Simpson]

Robert Tragesser posted two short essays

  "FOM: the Borel universe/but it wasn't blather -- a problem about
  meaning" (December 7)

and 

  "FOM: Meaning vs Significance Nailed Down, Work for Logic and
  Rev.Math." (December 10)

on "meaning" and "significance" as these concepts apply to CH.  Let me
present the perspective of one who, although deeply interested in
philosophy, is not an academic philosopher.

Preface: In another posting, Tragesser says:

 > Need there be a summary that would prove that it was all
 > worthwhile?  In any case, how could someone summing up guess the
 > stimulating, if kaleidoscopic, effect a thread might be having?

Against this rather Dionysian remark, I would like to note that
knowledge which cannot be summarized cannot be retained in conceptual
form.

In his first essay, Tragesser seems to be saying that the
"significance" of CH lies in the fact that we understand CH
informally, intuitively, in the light of nature, while the "meaning"
of CH lies in its status as a merely formal statement, subject to
various intuitive understandings.  For instance, according to
Tragesser_1, if we switch from ZFC-style sets to Borel sets, then CH
retains its "meaning" but changes its "significance".  I have two
comments: (1) This proposed distinction appears to be scientifically
invalid, since there is no reason to make formal statements before we
understand them to some extent.  (2) Even if the proposed distinction
makes sense in some unspecified context, I don't see that the proposed
use of the terms "meaning" and "significant" is consistent with any
accepted standard usage, as defined for instance in Tarski's treatise
on the semantics of predicate calculus, or in the Oxford English
Dictionary.  Therefore, I reject Tragesser_1's proposed usage.

In his second essay, Tragesser conflates the alleged distinction
between "meaning" and "significance" with a well-known, time-honored
mathematical distinction: elementary versus non-elementary proof.
According to Tragesser_2, "meaning" refers to the existence or
possibility of an elementary proof, while "significance" refers to
non-elementary proofs.  My comment on this would be that Tragesser_2
appears to be inconsistent not only with Tragesser_1 but also with the
accepted usage of "meaning" and "significance" as general
scientific/logical concepts.  Certainly the issue of elementary versus
non-elementary proof is important in mathematics, but so far as I know
it has never arisen in other disciplines.  And even if it has, the
transition from there to an alleged distinction between "meaning" and
"significance" strikes me as arbitrary.  Therefore, I reject the
Tragesser_2 proposal regarding "meaning" and "significance".

[ Having said that, I would comment that it may be interesting to try
to consider what an elementary proof of CH might look like, if there
could be such a thing.  But this seems extremely speculative. ]

In sum, it seems difficult or impossible to reconcile Tragesser's
various proposals with the way the terms "meaning" and "significance"
have been used elsewhere on the FOM list and, for that matter, with
general logical/scientific usage.  These proposals seem to exemplify
an academic impulse to redefine basic logical/scientific concepts
locally, in a way that may resonate with the whim of the moment but
which ignores the needs of science as a whole.  I'm curious as to
whether such proposals meet normal academic philosophy standards.
Could some kind person enlighten me, perhaps in private e-mail?

So far I have commented only on Tragesser's specific proposals, which
I reject.  Now let me make some positive remarks arising from other
parts of the CH thread.

1. When we are struggling with apparently difficult and speculative
f.o.m. issues such as the logical/scientific status of CH, it makes no
sense to tinker with very basic, very general, logical/scientific
concepts such as "meaning" and "significance".  It's much more
sensible and fruitful to analyze various possible approaches to CH in
their own terms.  Harvey's long posting of December 8 entitled "FOM:
On CH/1" does this admirably and thereby identifies a great many
interesting questions and potential avenues for progress.  My posting
of December 6 entitled "FOM: the Borel universe (a positive posting)"
is intended as another contribution in this vein.

2. Tragesser mentions several examples of non-elementary mathematical
proofs.  Several of these may lead to interesting foundational
research, quite apart from CH.  (1) The Hadamard non-elementary proof
of the prime number theorem can probably be done in WKL_0, since
elementary complex variable theory holds in that context.  Then a
conservation result for WKL_0 (see my paper on Hilbert's program)
would imply that the prime number theorem is provable primitive
recursively, without going through Erd"os-Selberg.  In this
number-theoretic context, it's not unreasonable to identify primitive
recursive provability with elementary provability.  (2) The
fundamental theorem of algebra looks like a good test case for
Harvey's project of working out real algebra and explicit quantifier
elimination in (systems conservative over) primitive recursive and
elementary recursive arithmetic.  (3) Hopf and Gelfand-Mazur: I don't
know, but probably something interesting could be said in the way of
reverse mathematics.  I would take a wild guess that Gelfand-Mazur is
equivalent to WKL_0 over RCA_0.

-- Steve Simpson