[FOM] What do we quantify over?
Dean.Buckner at btopenworld.com
Mon May 19 15:41:53 EDT 2003
Quantities, of course, the most fundamental being discrete quantities or
"numbers" or "collections" of things. So, our old friends Alice and Bob are
just a number of people. We can, as I have argued, simply write
(E X) X = Alice and Bob
We can also say these people have a number, of course (they are two).
Number in this sense is just an internal property of the collection. Given
that Alice really is a different person from Bob, i.e. that Alice and Bob
are not one, then Alice and Bob are two.
Another way of expressing discrete quantity is as the cardinal number
corresponding to the ordinal number of the last person in the collection.
So, if Alice is the first person, Bob is the second (or "other", which
simply means "second", as in "every other day"), and if Bob is the last
person in that collection (there is no one alse other than Alice and Bob),
and if the word "two" corresponds to "second", then the number of Alice and
Bob is two.
Logically, we can do without cardinal numbers altogether. To specify the
number of a collection, we simply have to enumerate its members. So, if a
collection consists of the first object, a second object different from the
first, and a third object different from the first and second, and if there
are no other objects to be enumerated, then we have already specified how
many objects there are. The order in which I enumerate the objects is
unimportant. If I say that A is the first object, that B is the second
(i.e. it is different from A), and that C is an object different from A and
B, then I have expressed exactly the same proposition as if I had said C is
the first, B the second, A the third. In both cases I have said that each
object is different from each of the others.
A is different from B, C and D, B from C and D, and C from D
says that there are four objects and so on. We can reduce statements of
number to statements that presume only the concepts of difference, and of
Statements like "there are a number of objects" or "there are n objects"
state the existence of a number of things without specifying what their
cardinality is. We can only make the cardinality explicit using an identity
S = a1, a2, a3, a4.
where "a1", "a2", "a3", "a4" are singular terms referring to each of the
objects in the collection. But note, this is an identity statement. We are
using one term "S" to refer to a bunch of objects, and saying of them that
they are identical to the bunch of objects referred to by "a1, a2, a3, a4",
and in doing so make the cardinality of what is referred to explicit. (Just
as the statement "Cicero = Tully" is a covert statement of number).
When is the cardinal number of two collections equal? When the propositions
expressed by enumerating the objects are the same. If for example I say
"this (of A) is one thing, this (of B) is a second", I have said that there
is one thing, and that there is another. And if I say "this (of C) is one
thing, this (of D) is a second", I have ALSO said that there is one thing,
and that there is another, I have expressed the same proposition (my two
sentences have the same meaning).
It's implicit in our whole concept of enumeration - giving separate proper
names ("A", "B" ...") to a bunch of things for which we only had a single
proper name ("those things") that we give the number of the things
enumerated. It's also clear that this gives us only the concept of finite
numbers of things (since we can reduce the concept of cardinality right down
to the repeated assertion of difference). So, when we quantify over
discrete quantities like this, using concepts that are devoid of any
set-theoretical content, we quantify over finite things.
There's more to say about continuous quantities, which we can also quantify
over, but I'll stop here.
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