FOM: Numbers of objects

Dean Buckner Dean.Buckner at
Sat Apr 20 07:32:19 EDT 2002

Robert Williams  asks which bit of Fregean dogma I really take issue with.
Briefly, it is the idea that "a statement of number ascribes a number to a
concept".  Number in my view is not something that belongs to concepts, but
to objects.  It's a property of a collection, or a "number" of objects.

Some more arguments for this idea:

1.  The predicate "x is a different thing from y" is a 2-place predicate,
satisified by all couples of things (Caesar and Anthony, Anthony and
Cleopatra, Caesar and Cleopatra ...).  Its terms are singular ("Cleopatra",
"Caesar" ...) and thus signify individual things, not concepts.  Obviously
these may be abstract things, such as in "love is a different thing from
hate", but that's not what Williams has in mind. Is it?

2.  A constantly recurring argument given by Williams and others (and from
Frege before he saw the light in the early 1920's) ) is that any expression
of the form "the F", has to designate something.  What does "the largest
prime number" designate?

3.  On the argument that we can define "2" as the set of objects
equinumerous with {0,1}.  This presupposes that "0" and "1" designate
anything - where exactly is the argument for this?  Frege has this idea that
each number is associated with an "extension" but I admit I have never
understood it.  An extension in classical (pre-Fregean) logic is just the
objects that fall under a concept (predicate).  But since Frege has to start
with the concept "non-self-identical", which doesn't have an extension, I
don't see how the argument gets off the ground.

Other people use the idea of an "empty set" but I don't understand that
either.  We used to have a nice set of four crystal glasses  which
unfortunately I dropped on the floor.  So we don't have a set of glasses any
more.  Or do we still have an empty set?  What?

It also presupposes 0 is a different thing from 1.  Otherwise we would end
up with the set of objects equinumerous with {Cicero, Tully).  In other
words, we presuppose that 0 and 1 themselves satisfy the predicate "x is a
different thing from y".  Can we define equinumerousity with {0,1} in a way
that doesn't presuppose the use of this predicate?  Why would we want to?


In summary: "x is a different thing from y" is a first, not a second level
concept that defines all collections of two objects.  It suffices for the
definition of number, and without any vacuous nonsense like empty sets or

Dean Buckner
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