FOM: Reply to Kanovei on counting

[Neil Tennant]

On Fri Oct  9 09:48 EDT 1998, V.Kanovei wrote:

> There is a point not properly attended by NT. 
> He basically argues that 
> 1) Measuring angles with enough precis[ion] 
> 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, ...

Actually, I never argued to that effect. I would be happy to leave the
measuring problem to the physicists who need to find whatever
empirical information (within available tolerances of measurement)
they regard as necessary and, for the time being, sufficient, in order
to make the best conjecture possible as to what "the" correct geometry
for physical space might be.

Kanovei continues:

> 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 opposit[ion] to 1) above (on geometry).

Actually, I did not say anything about counting, either; although I'd
be prepared to say something if pressed. First, every result reached
by counting "the physical objects in" a finite collection C will be
*exact* in a rather trivial sense: the result will be a particular
natural number. Kanovei's worry must then take the form: Is this
natural number n that we have reached (as the result of our attempted
count) the true measure of the size of this finite set?

In one obvious sense, one can always answer "Yes", EVEN IF

> objects in C will necessarily appear or disappear 
> with some non-0 probability, so ... is it at all 
> physically sound to claim that a big enough C 
> has a certain number of objects ? 

Note that (provided we don't ever count the same object twice) we can
make the contingent claim

(1)	The C-type objects that we have counted form a collection of
	size n.

This is equivalent to

(2)	The set {x|x is a C-type object that had a natural number
	assigned to it in the course of our counting how many C-type
	objects there are} has cardinality n

and to

(3)	#x{x is a C-type object dealt with in our count} = n*

where n* is the numeral for n.

	The latter is of course the left hand side of an instance of
the a priori conceptual control given by Schema N:

(N)	#xF(x) = n* iff there are exactly n F's

Note that none of the equivalent claims (1), (2) or (3) is the claim
that *the collection C* has size n!  For the problem is, ex hypothesi
Kanovei, that C is both (with non-zero probability) "losing and
gaining members". Since that is the case, there is no unique
collection C, over time, that we are talking about here. Rather, there
is a unique (temporally indexed) *concept*, perhaps ("...is a C-type
thing at time t"), whose extension is fixed only if t is fixed. But the
time-dependent extension of this concept can change discontinuously
from one moment to the next. Each time a C-type thing pops in or out
of physical existence, a *different* collection of C-type things
results. The problem is simply that actual counts take time, while the
extension of a concept can dependent on time. But this kind of problem
faces us all the time---for example, in conducting a census even in a
small nation like the Vatican. Indeed, given geriatric hostage to
fortune, even an answer to the question

	"What, right now, is #x(x is a Cardinal in Rome)?"

has to be taken with fingers crossed.  Why should we have to go
quantum-mechanical to skin this cat?

None of these reflections about the fickleness of physical existence
can impugn Schema N. Rather, adherence to Schema N is the pivot around
which other reflections fall into a coherent picture. The a prior
control exerted by Schema N over all our thinking about things and
their numbers (under different concepts) is completely unyielding.

Moreover, Schema N would govern the thought of any rational creature
WHATEVER the nature of the physical universe that it was forced to
inhabit. Even if I were a disembodied Cartesian ego, the sole
contingent existent in the universe, I could proceed in thought as
follows:

	#x(~x=x)	= 0
	#x(x=0)		= s0
	#x(x=0 v x=s0)	= ss0
	:
	#x(x=0 v ... v x=n*) = sn*
	:

and I could work out, in purely logical fashion, the Peano-Dedekind
axioms. I could also derive all instances of Schema N.

I could never, however, do that for the geometry of physical space,
were there to be contingently existing bodies needing a space to
inhabit, in order to render my empirical experience of them possible.
 
Hence, a propos the fundamental point at issue---that arithmetic and
physical geometry really are very different indeed in their respective
epistemological status---Kanovei's worry about physical objects being
of fickle constitution is at most a trifling, and temporary,
distraction. We are dealing with *a priori* matters here, when
determining how and why thought about natural numbers turns out to be
applicable to reality (both abstract and concrete).

By contrast, in the case of physical geometry, we are dealing with *a
posteriori* matters when identifying geodesics,the metric, curvature
etc. Except, perhaps, for one aspect?---namely, the possibly a priori
status of the *topological* features of any space that could be the
space of a rational creature's empirical experience. That was Carnap's
own neo-Kantian contention, at any rate, at the time of his
Doktorarbeit and before being Vienna'd by the time of his
Habilitationsschrift.

Neil Tennant