FOM: The black and white cats (Frege #2)

[Dean Buckner]

This is the second in a series of short objections to Frege's arguments
about
number.

Incidentally, offline correspondence suggests people don't have any
counter-objections to what I argued in #1 and elsewhere.  It would be useful
to know if evereone agrees.  If so, I see little point in further postings
on this subject.  But people shouldn't agree!  The Fregean arguments I am
attacking are historically important to the development of set theory (as we
know know it), otherwise Frege wouldn't have spent so much time and energy
on them, in Grundlagen and elsewhere.

                                        *******

#2 The black and white cats

In a review of Husserl's Philosophy of Arithmetic, written in 1894, Frege
criticises Husserl's view that our abstract idea of a collection or set is
formed by not "specially" attending to something.  The criticisms were
instrumental in converting Husserl away from the psychologism espoused in
that work (& the rest is history)

He acidly remarks that such inattention "must be applied at not too great a
concentration, so that everything does not dissolve, and likewise not too
dilute, so that it effects a sufficient change in things.  Thus it is a
question of getting the right degree of dilution; this is difficult to
manage, and I at any rate have never succeeded".

For example, if we attend to the blackness and whiteness of two cats, our
inattention is too dilute for this does not give us the abstract idea of
"two".   Likewise, we must somehow ignore their posture, position, and
catness. But once we do, we get just a "bloodless phantom", a *something*
that is somehow different, in  way we cannot easily specify, from a
*something* obtained from the other object.  We have dissolved everything!
If we abstract too little, we get things that have different properties.  If
too much, with things that are the same.  Either way we do not get the idea
of abstract numerical difference.

This is a very effective piece of writing.  But as usual, Frege hits the
mark, while aiming at entirely the wrong target.  If he is entirely correct,
then we cannot possibly grasp the sense of the proposition

Something is different from something

But we can.  The proposition is true when something - no matter what - is
different from something else - no matter what - and that's how we grasp it.
It consists of two terms, the "somethings", and the relational predicate "x
is different from y".  The relation is symmetrical.  Why should we expect
the abstract meaning of one "something" to be different from the other?  It
is not there that we should look for the abstract meaning of "different
from", but in the predicate itself.  As Frege himself says (Geach & Black p.
79): though
we cannot have another person's mental image, it is "quite otherwise" for
thoughts.  "One and the same thought may be grasped by many men".  Your idea
of difference may be a black and white cat, mine may be a large and a small
one.  But the meaning of "x is different from y" must, if Frege is right, be
the same for everyone!

The mistake consists in treating meanings as objects.  If the meaning of the
first "something" is indistinguishable in every respect from that of the
second "something", then the two meaning-objects must be identical.  And if
"x is different from y" is predicated of the meanings, then the proposition
is necessarily false.  But "something is different from something" does not
say that the meaning of "something" is a different thing from the meaning of
"something".  It says, there exist two things - we don't know which - are
different.  (The proposition might, if the world consisted just of two
meanings floating around in a disembodied and otherwise vacant mind, happen
to be true of two meanings, but they would still not be the meaning of
"something").

Frege is thus victim to a naïve fallacy that he exposes elsewhere  (when we
say "the moon", we do not intend to speak of our idea of the moon, or we
would have used the phrase "my idea of the moon" (On Sense and Reference,
Geach & Black p. 61)

                                *****

May I commend a most excellent piece by William Tait.  It has a dull title
("Frege versus Cantor and Dedekind: On the Concept of Number"), but
interesting content, including similar argument made by Frege on Cantor's
theory of abstraction.  See secs 8-9.  The paper is on Tait's website, I
have no address at hand but elementary search should reveal.



Dean Buckner
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