FOM: Arithmetic vs Geometry : Categoricity

[Kanovei]

There is a point not properly attended by NT. 
He basically argues that 
1) Measuring angles with enough precise 
we shall always have some small \epsilon of 
difference from the exact \pi=180, 
hence there is a problem to find that theoretical 
geometry which explains this \epsilon most 
effectively, 
2) Counting objects, we always have an exact result 
which does not depend on the type of objects, their, say, 
colour or shape, do we use computer or, say, fingers 
to count, et cetera. 
In other words, physical counting is claimed to be 
in 1-1 precise correspondence with mathematical 
counting, in opposite to 1) above (on geometry).

However is 2) that true ? 
Let's count a collection C of enough many objects 
using a counting device D. By Quantum Mechanics, 
objects in C will necessarily appear or disappear 
with some non-0 probability, and D will have 
similar problems, so, basically, is it at all 
physically sound to claim that a big enough C 
has a certain number of objects ? 
If it is not then 2) becomes questionable. 

V.Kanovei