higher mathematics of the 1800s had proceeded by exploring successively more
abstract levels of the function. Frege’s stroke was to recognize that the central
tool of modern mathematics could be extended to logic.
Frege starts not with concepts and what is predicated about them but with
judgments. His key idea is that the step from a judgment or assertion to a
concept is analogous to the mathematical relationship between a function and
its variables. Instead of assuming that we already know the concept, Frege
begins with it as an empty frame to be filled.^7 From this perspective singular
and general representations are entirely different kinds of things, arrived at by
very different procedures: the singular (“this horse”) is a proper name given
independently of judgment, while the general (“horse,” “horses”) arises only
after a judgment has been made.
In Frege’s vision, talking about the world does not consist in making
connections, so to speak, horizontally on a plane; it is a hierarchy of levels.
Making use of the technical tool by which the machinery of modern mathe-
matics was created, Frege introduces a formal symbolism to force automatic
recognition of new conceptual distinctions: using different print fonts to sepa-
rate clearly use and mention; a symbol indicating that something is asserted,
distinguished from what is asserted; strokes which replace “not,” “and,” “or,”
“if then”; and the universal quantifier “for every value of x,” which renders
the distinctions among the “all,” “some,” and “none” of ordinary language.
Confusions in the older rhetoric now come into view. To speak of a “quality”
had confused the unasserted content with the fact of its being asserted (Coffa,
1991: 63). The copula is not something separate which links a subject and its
qualities, but an aspect of the functional statement; for this reason, Frege
([1883] 1980: 65) comments, the ontological proof of the existence of God
breaks down, since existence is not a quality.^8
Frege came upon the issue by way of clarifying the concept of number.
Ordinarily “one” and “unit” are taken as synonymous. If we count three
objects (1 1 1 3), how is it possible that objects which are different
can all be treated as identical?^9 The number 3 is not an agglomeration of objects
collected together, since they retain their properties which made them distinct;
but if we are counting identicals, we never reach a plurality (Frege [1883] 1980:
50). The mathematical plus symbol () cannot be interpreted as the “and” of
ordinary language. The solution is to recognize number as a self-subsistent
object. Frege considers it the extension of a concept, that is, the set of all
instances which fall under that concept. Frege’s numbers are Platonic; they are
not derived from counting or sequences, nor are they properties abstracted
from things in the sense that colors are. This shifts our attention to the
procedure by which we make assertions about numbers, to the judgment that
702 •^ Intellectual Communities: Western Paths