[FOM] Platonism and Formalism
Haim Gaifman
hg17 at columbia.edu
Thu Sep 25 18:48:08 EDT 2003
>Harvey Friedman Wrote:
A specific assertion first order assertion phi in the finite structure
(V(9),epsilon)
where V(9) is the 8th level of the cumulative hierarchy starting with
V(0) =
emptyset, will be found with the following properties:
i) phi is simple and clear and natural;
ii) phi is generally regarded as a noteworthy mathematical truth if
true;
iii) many close variants of phi are equally noteworthy and published
proofs/refutations of them by attractive standard methods appear,
causing
there to be an attractive field of research surrounding statements
like phi;
iv) phi and its close variants make just as good sense in V(omega) as
they
do in V(9), with the situation in i),ii),iii) prevailing:
v) a proof of phi is published which is carried out within ZF + "there
exists a nontrivial elementary embedding from some V(kappa) into
V(kappa),
where kappa is an inaccessible limit of inaccessible cardinals";
vi) a proof is also published of the following metamathematical
result,
proved using only the basic properties of V(11) - i.e., the proof is
"carried out" within (V(11),epsilon), but itself appears as 100 pages
in the
Annals of Mathematics.
*If phi is true then there is no proof of phi in the system ZF +
"there
exists a nontrivial elementary embedding from some V(kappa) into
V(kappa),
where kappa is an inaccessible cardinal", which is small enough to be
an
element of V(7).*
-------------------------------------------------------------
Dear Harvey,
First, an apology to you and the FOM
list for the very late reply.
I have had virus related problems and
a very big backlog to take care of. The points I am going to make
might have been touched upon by other messages
I have not seen, but I shall make it anyway.
I am delighted by your message and hope
that you shall have soon the full proof of
the result you claim. (I am not sure I understand
the passage about about a forthcoming
100 pages proof in the Annals of Math, but this does not
matter).
But I cannot see why you think that this
should make Platonists uncomfortable.
On the contrary.
Platonists claim that there are objective
mathematical facts, but they do NOT claim
that such facts (or such interesting facts) are knowable
to us.
The result you predict shows clearly
that if we believe that every statement is either
true or false in V(9), then we should also accept that
certain facts are inaccessible to our knowledge.
This situation strengthens the view that mathematical truths
as *facts*, whose status does not derive from
our ability to prove them, or verify them in this
way or another. So either one is a strict finitist
that does not believe in the meaningfulness of iterating
the power set operation 7 times 2^(2^(^...(^2)...), or one is
a "minimal Platonist", that is: a Platonist with respect
to very large finite universes. Recently I made this very same
point in my talk in Oviedo (The History, Philosophy and Methodology of
Science Congress), and I cited in support of it results about lower bounds
in complexity theory. Your message provides a more
concrete and dramatic way of supporting this point.
It is true that the independence results in set theory have
given trouble to Platonists. But this is due to the
way they have been proven, rather than the results themselves.
The methods of manipulating the semantics and of
getting the multidue of models, in which the power set
behaves in completely different ways, indicates
the possibility that the general concept
of set might be ambigous.
An independence result, in itself, need not weaken a
Platonist view. If it concern an area about which we
have very firm intuitions, e.g., V(9), it strengthens our
view of mathematics as being about an objective reality,
whose truths might remain unknown to us, because
of out limitations.
But if it concerns an area in which
we feel shaky to start with, it might raise doubts about
the objective reality of the domain. Still a lot will depend on the
way the result is proven.
Haim Gaifman
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