FOM: The foundational relevance of theology

[Joe Shipman]

Simpson:
The theology thread has gotten off-topic.  Could I ask everyone to
please make sure that your future FOM postings have some substantial
f.o.m. content?  Experience has shown that postings with no
f.o.m. content can do serious damage to the FOM list.

The posts by Holmes, Tennant, Ketland, and myself have serious f.o.m.
content, albeit sometimes amidst non-f.o.m. content.  Please don't shut
down the discussion prematurely.

The major issues are twofold: ontological and epistemological.

On the ontological side, the radical finitist position is clearly at
stake.  Ignoring theological debates about the more subtle attributes of
God, there is general agreement that God has (or rather that anything
satisfying the predicate "God" must have, which leaves open the
possibility that nothing satisfies this predicate) the attribute of
infinity.  Furthermore, it is a common tenet of many theological
traditions that God has the attribute of self-knowledge, which suggests
that God knows whether there are infinitely many twin primes (since as
an infinite being God can be regarded as containing or instantiating at
least the smallest kind of infinite set, which can be represented by the
integers), and the contention that such statements need not have a truth
value becomes dubious.

Ketland already accepts even uncountable sets because they are
(apparently) necessary for our best scientific theories.  There was an
interesting discussion here about this earlier this year.  Therefore the
simple assumption of God's existence does nothing for his mathematical
ontology.  On the other hand, a more Cantorian notion of God would give
ontological support to much larger infinities, for example proper
classes.  Rudy Rucker's book "Infinity and the Mind" contains some good
speculation along these lines.

The epistemological question has been addressed by Holmes.  Here the
putative benevolence of God is relevant: not only negatively (that God
won't confuse us) but positively (the world is knowable, either by
rational investigation, by divine revelation, or in Heaven).  Tennant
has an interesting point comparing God to other not-directly-observable
entities whose existence we are comfortable in postulating because they
give conservative extensions; here the theological issue of God's
continuing activity in the world becomes relevant (only an aloof,
Deistic God would conservatively extend physics).

Epistemelogically, I don't think the concept of proof is affected by
theology, but the intuition that supports our acceptance of axioms as
true may be regarded as God-given.

Can anyone document Godel's view on these matters?  In particular, did
his view of mathematical intuition involve any concept of God, and what
motivated his "ontological proof"?

-- Joe Shipman