the number of the wineglasses on the table is four—a statement of identity as
to which number that is.
The procedure for establishing numerical identity is independent of count-
ing; it is to establish a one-to-one correlation between the objects in each set,
just as a waiter need not count all the silverware but only lay one fork alongside
every knife (Kneale and Kneale, 1984: 461). Frege goes on to build the entire
number system by purely logical definitions. Zero he defines as the set of all
objects which are not identical with themselves; this is a logical impossibility,
so nothing falls under it (a fateful move in view of Russell’s later paradox).
The category zero is absolutely simple in Frege’s assumption, so now he can
define “1” as the set of all sets which are identical with the zero set (not with
the contents of the zero set). Further numbers are built up as sets which contain
all of the preceding number sets (2 is [zero, 1]; 3 is [zero, 1, 2]; etc.). Paradoxes
about infinite numbers, just then giving rise to scandal because of Cantor’s
unending series of transfinite numbers, are taken care of in Frege’s nested levels.
The number belonging to the concept “finite number” is an infinite number;
it is not a number following in the series of natural numbers. Frege allied
himself with Cantor, who had begun his theory of sets only a few years earlier
in 1874.^10 This was an alliance of the radicals against the establishment
controlled from Berlin by Kronecker, the rigorist enemy of paradox-generating
abstract methods in mathematics.
In 1892 Frege introduced a distinction between sense and reference. The
problem arises in interpreting the equal sign () in a mathematical equation.
If this is strictly a sign of identity, then 1 3 4 can be replaced with 4
4, an uninformative statement. The equation is telling us something, but not
about the referent of either side of the equation; that referent is the number-
object 4 in either case. Each side has a sense as well as a reference; the sense
of 1 3 is different from the sense of 2 2. The same can be said of verbal
expressions: “the morning star is the same as the evening star” is uninformative
insofar as both parts refer to the planet Venus; but each expression arises in a
different semantic context and has a different sense. A referent is an object
(which for Frege is not just something perceptible, but can include numbers,
times, and so on). The sense is on a different plane, the semantic means by
which referents are singled out for attention (Kneale and Kneale, 1984: 496;
Wedberg, 1984: 113–122). Propositions as well as names and expressions have
referents; for Frege, the referent of a proposition is its truth value. Thus all
true propositions have the same referent, the True, just as all false propositions
refer to the False.^11 Frege’s sense-reference distinction was not picked up until
the 1920s, when Carnap began to use it in a strongly reductive program,
counting propositions as scientific only if every name in them has not only
The Post-revolutionary Condition^ •^703