[FOM] inconsistency of P

[Timothy Y. Chow]

On Tue, 4 Oct 2011, Arnold Neumaier wrote:
> Or is there a doubt mapping that assigns to each natural number a doubt 
> value, such that small natural numbers have doubt value zero, and the 
> doubt increases to large values if numbers get really huge? Would this 
> doubt function depend on who doubts, and thus be subjective? Or is it a 
> property of the universe only? Neither option looks very convincing....

Convincing as what?  As a nicer and more elegant theory of the integers?  
Then I agree with you.  Or as a more accurate description of what natural 
numbers "really are"?  Then I think the question is debatable.  If we 
posit that the existence of natural numbers is somehow intimately tied up 
with the physical world, it should not be surprising that the harder we 
try to align our theory with reality, the messier it gets, with sorites 
paradoxes creeping in and so forth.

In a first course in quantum mechanics, we're taught to represent various 
subatomic quantities as wave functions.  Is an electron *really* a wave 
function?  Everett might say yes, but others woud say no: it's a useful 
fiction that is theoretically tractable and helps us make experimental 
predictions.  Similarly, in general relativity, we represent spacetime as 
a pseudo-Riemannian manifold.  Is spacetime *really* a pseudo-Riemannian 
manifold?  Is the continuum hypothesis a physical question whose truth 
depends on whether there is really an uncountable set of spacetime points 
out there not in one-to-one correspondence with the set of all spacetime 
points?  Most would say no.  The physical world is a messy place and if we 
want as *accurate* as possible a description of it, then our beautiful 
theories are not up to the task.  This is not to say that we shouldn't 
create and study beautiful theories, but we shouldn't conflate our 
beautiful fictional stories with the ugly reality that they are trying to 
approximate.

Tim