FOM: more on God, numbers etc.
neilt at mercutio.cohums.ohio-state.edu
Tue Oct 19 23:33:08 EDT 1999
In my posting about the `Ontological' Argument for the Non-Existence
of Zero, I forgot to take the opportunity to note that, while Anselm's
original argument was strictly classical (inferring the existence of
God by reducing to absurdity the assumption that God does not exist),
the spoof argument about Zero is completely intuitionistic (inferring
the non-existence of Zero by reducing to absurdity the assumption that
Zero does exist).
Those who accept BOTH arguments will have to face what might be called
Kronecker's problem. Remember that Kronecker said "God gave us the
integers; Man did all the rest." Kronecker would have to accept God on
non-constructive grounds, and then deny the existence of God's gift to
Man, for want of Zero!
In a slightly more serious vein, I'd like to note an interesting
tension that could arise in the thinking of an atheist
mathematician. It is common for confidence in an "ontological
extension" to be secured by a conservativeness theorem. That is, the
newly postulated entities don't seem so dodgy if one can show that, by
thinking about them, one cannot prove any new results about the old
entities that one hadn't been able to prove before. The extended
system is therefore consistent relative to the old one; and the
ontology of the extended system is thereby thought to be a little more
But, when it comes to undermining belief in God's existence, the
rational atheist (such as Jeff Ketland) tends to take the Quinean line
that we should acknowledge the existence only of such things as have
to be quantified over in our best scientific theories. The tacit
assumption is that postulating the existence of God, or of gods,
angels, devils and the like would secure at best a conservative
extension of the God-free scientific theorizing that we prize so
highly. That is, we do not need to posit God in order to make all the
predictions we wish to make about the natural order.
If the theological extension of natural science is a conservative
extension (with respect to our observation sentences), why does our
confidence in physical posits not then transfer to the divine posits?
Does this situation deserve some such label as "methodological
For my own part, I think not. But it is the Quinean who would be in
difficulties in the claimed absence of any way of distinguishing those
sentences that are cognitively significant from those that are
not. Without a criterion of cognitive significance, the Quinean might
not be able to weed out God-talk from a theory dealing variously with
God, physical things, and mathematical entities.
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