[FOM] Falsify Platonism?

[Brian Hart]

I have a suggestion which may not be a proof or disproof of Platonism
but it could be thought of as yet another pragmatic justfication.

Impredicative methods seem to be gaining ground in mathematics and
logic.  Friedman's current research program to find natural
applications of LCAs is an important family of related results
frequently making use of impredicative LCAs.  Kruskal's theorem is
another important use of mathematical impredicativity.  And there are
many other examples.  And after a certain threshold of largeness (can
anyone specify it?) most of V requires impredicative definability,
thus if our confidence in the legitimacy of the enlargement of V
continues unabated then so should our confidence in the legitimacy of
the use of impredicativity since it will become increasingly
indispensable.  Keep in mind that mathematical certainties need not be
equivalent and so we may never become as epistemologically confident
in impredicative methods as we are in their predicative relatives, but
as one undertakes the Pythagorean ascension from the lower to the
upper echelons of mathematical existence one's confidence need not
approach zero but can still said to be decreasing.

If one axiomatizes the logical universe (the one containing strictly
logical objects such as proper and hyper-classes) impredicativity is a
requirement as these objects cannot be defined non-circularly.

Even an axiomatization of certain physical theories may require
impredicativity.  For example, as Hellman points out in his paper
"Beyond Definitionism -- but not too far beyond", Quantum Field Theory
(QFT) requires the impredicative use of non-separable Banach and
Hilbert spaces.  And further examples of physical impredicativity may
be discovered soon.

Axiomatizing physics as a whole may too require impredicativity if one
considers the physical universe to be an objective domain (as the
common sensical and scientifically realist viewpoints agree upon).
Thus, perhaps impredicativity is not just significant for objects but
for objective domains as well.  The physical universe itself is not
usually thought of as a single object, but perhaps it is a kind of
united whole according to the principle of Platonic One-ness.  Perhaps
the principled nonseparability of quantum systems is also a kind of
manifestation of the unity of Being.  In contrast, classical systems
may be separable but they're still not causally dissociated from the
influence of external forces.  Love may be another such manifestation
-- even the social order prefers unity.  Sure, the greatest of
emotions may be genetically encoded but its primordial origination may
be metaphysical in nature.  This is consistent with Plato's view of
love as a Platonic Form.  As for the other two domains (besides the
logical, mathematical and physical ones) of Plato's penta-domain,
namely the domains of the Beautiful (B) and that of the Ideas (C) (or
concepts in less Platonic parlance), both would seem to require
impredicative foundations since, following the conceptual realist
Gödel, all possible concepts (at least the objective ones) exist
objectively.  For the rest of this paragraph assume V is a proper
class, then so should its associated class of mathematical concepts,
for each statement of mathematics should require at least one concept
(representative of a single mathematical object), should it not?  The
minimal mathematical statement should be a single, simplest object, or
one referring to itself (perhaps via an automorphism) if one requires
relationality.  A predicative minimal statement must require a single
predicate to assign the object a single property.  As for B, if all
mathematical objects are abstractly beautiful then this serves as a
lower bound of the logical magnitude of a sub-domain of B, thus it too
is an impredicative totality.  Thus, in this speculative Platonic
picture, impredicative foundations would seem to be the norm rather
than the exception which is converse to its present acceptability.

Surprisingly enough, impredicativity can even occur in a context as
presumably predicative as the inductive definition of a computable
function of finite type, since it is defined circularly, as mentioned
on pg. 73 of Gödel's Collected Correspondences (H - G).  He even tried
to remove this impredicativity in his Dialectica paper but had a
difficult time due to his uncertainty as to the exact bound of
finitary reasoning and this significantly delayed the paper's
completion.

Thus, if there _is_ objective existence, it would seem impredicative
definability is a necessity.