between sense and reference. In this spirit Frege undertook a reform in the
entire system of notation and operations which made up traditional logic,
replacing the taken-for-granted concepts of subjects and predicates, modeled
on language, with a formal system which makes explicit the multiple dimen-
sions of operations brought out in the mathematics of numbers. It was this
machinery which Russell saw could be as revolutionary in the realm of phi-
losophy as was the machinery of equations with which Descartes had promoted
the takeoff of modern mathematics.
Once again, the straightforward path toward solving all problems with a
new tool became displaced into a fanning-out of creativity at the newly exposed
ground level. What made the Frege-Russell movement philosophically fruitful
was not so much its avowed intention, the technical exercise of translating
philosophical problems into the new formalism and thereby clearing away
confusion forever, as that it uncovered deep troubles. Russell became more
famous for his theory of types, his device for repairing a deep trouble, than
for his logicist program itself. The real fame of the movement came with
Wittgenstein, who built a meta-system by wrestling with the general difficulties
in hierarchical programs such as the theory of types, and with the Vienna
Circle, who promoted Russellian logicism as the revolution which was to sweep
away all the fallacies of philosophy. After the fireworks of the initial phases of
combat, what settled out as the new philosophical turf was the exploration of
what kinds of things are sayable in various sorts of systems of notation or
symbolism. In a sense Wittgenstein amplified Kant’s discovery about the syn-
thetic a priori: what had previously been taken as merely arbitrary and tauto-
logical symbols turned out to be a realm with its own contours, an unknown
land to be explored, and in which discoveries might be made far beyond the
taken-for-granted of past history or “common” sense. This is much the same
as what higher mathematics does, in full reflexive consciousness; this is why
we can say that the mathematical abstraction-reflexivity sequence has been
the instigator of the philosophical abstraction-reflexivity sequence in modern
Europe.oppositions of higher reflexivity: formalism and
anti-formalism to husserl and wittgenstein
A large segment of modern European philosophers would have been surprised
to hear mathematics described as the driving force of modern philosophy.
Nevertheless, the intellectual world is structured by oppositions; there is not
merely a non-mathematical philosophy in the modern attention space but a
wing of militantly anti-mathematical philosophers. The weapons of modern
anti-mathematical philosophies have owed more to their enemies than the
anchorage which enemies always provide for one another. The reflexivity of854 •^ Meta-reflections