[FOM] The boundary of objective mathematics
joeshipman@aol.com
joeshipman at aol.com
Thu Mar 12 11:22:41 EDT 2009
-----Original Message-----
>From: Paul Budnik <paul at mtnmath.com>
>I an dividing mathematics into absolute objective statements and
>relative ones that are sometimes treated as if they were absolute.
....
>The difference I want to make is between an an arbitrary path that is
>followed by a recursive process in a potentially infinite universe and
a
>completed infinite set. This is a distinction that goes back at
least
>to Aristotle. My position is perhaps close to constructivists, but I
do
>not demand a constructive proof of a statement. I only demand a
>constructive proof that all the events that determine the statement
are
>themselves determined by finite events.
The practical attitude many mathematicians seem to take is that
statements in the first-order language of arithmetic OR statements of
higher type which have arithmetical consequences are meaningful. For
example, Schanuel's conjecture cannot be formulated in arithmetic but
using it one can prove that (e^(e^n)) is never an integer when n is an
integer, a statement which can be given an arithmetical formulation in
terms of convergence of computations. The existence of a (countably
additive) real-valued measure on all subsets of the continuum is a
statement of even higher type which has useful arithmetical
consequences such as Con(ZFC).
Many mathematicians would also declare as meaningful statements those
which are set-theoretically absolute (have the same truth value in all
transitive models of ZFC). For example, the Invariant Subspace
Conjecture (all bounded linear operators on Hilbert space have
nontrivial invariant subspaces) does not have any arithmetical
consequences that I know of, but is considered to be one of the major
open problems in mathematics. (Of course one can't prove that the
Invariant Subspace conjecture has no arithmetical consequences without
proving it consistent, which might be no easier than proving it
outright.)
My own view is that any statement about sets of bounded rank is
meaningful, and that statements like GCH which involve universal
quantification for sets of arbitrary rank are vague. In between these
two classes of statements are existential statements with no bound upon
the rank -- statements like "a measurable cardinal exists". (In other
words, the statement "no measurable cardinal exists" is vague while "a
measurable cardinal exists" is less so; this asymmetry is not
unreasonable and is analogous the asymmetry between the statements "the
Riemann Hypothesis is not provable" and "the Riemann Hypothesis is
provable" which have different epistemological statuses).
-- JS
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