Dummett's Backward Road to Frege and to Intuitionism - Tripod

well-defined as mutual and is therefore one-one (Frege 1964: 92–94). We may say that functions which
have the same course-of-values are representatively identical. Representatively identical functions are
not different functions, though due to the “peculiarity” of predicative language we cannot directly say
so (Frege 1970d: 46). Since functions are incomplete, they cannot directly stand in the identity relation.
A function cannot even be directly said to be identical with itself. For “in view of its predicative nature,
it must first be...represented by an object” (Frege 1970d: 46). It is the representing object that stands in
the identity relation. We representatively say that functions are identical when we say that their
representing objects are identical. Since mutual representation would be impossible if there were not a
one-one correspondence between functions and their courses-of-values, the representative identity
conditions of functions are exactly as sharp as the identity conditions of the objects that represent them.
That this one-one correspondence obtains is the famous extensionality thesis (Furth 1964: xl–xliv),
whose name we may honor by saying that for Frege, functions are always extensional in sense (2). A
function is definable as any equivalent function (Frege 1970c: 80). Functions are equivalent if and only
if their courses-of-values are identical (Frege 1964: 43–44). Thus equivalence is the relation of
representative identity. The equivalence relation and the identity relation represent each other, as do
their respective relata. But the identity relation itself is indefinable, for a technical reason: since a
definition is a stipulated identity, all definitions presuppose identity (Frege, 1970c: 80–81); a definition
of a function is a stipulated representative identity. Indeed, the best evidence that representatively
identical functions are not to be regarded as different functions is Frege’s explanation (not: definition)
of identity as being indiscernibility (Frege 1970c: 80).
When Dummett contrasts intensionality with truth-functionality, he has sense (1) implicitly in
mind. When he discusses senses as different ways of presenting one object, he has sense (2) implicitly
in mind. And when he discusses our inability to traverse infinitely many objects which might fall under
a concept, he has sense (3) implicitly in mind. But I cannot recall that he ever expressly distinguishes
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