-An Excerpt from Christopher Norris' Badiou’s Being and Event, pgs. 52-55.
In effect, the very confidence initially displayed by Frege and Russell in the power of their logical language (technically speaking: that of the first-order quantified predicate calculus) to offer a complete and perfectly consistent formalization of the set-theoretical domain was itself a sure sign of the project’s being headed for just such an obstacle somewhere along the way. It is precisely this recurrent gesture of containment—this move to control and delimit the scope of enquiry through various techniques of always premature ‘totalization’—that Badiou regards as having posed a chief obstacle to progress by evading the radical or, indeed, socio-political order. Hence his objection not only to the claim that intuition might yield valid insights or conceptual progress (since intuition is most often just the name applied to preconceived belief) but also to that narrowly logicist idea that the exploratory scope of set theory might be circumscribed by a purely formal programme whose terms are specifically in advance and which therefore pre-emptively restricts any future developments to what is conceivable at present. So it was, Badiou thinks, that the Frege-Russell project was predestined to run aground on those paradoxes that sprang to view as soon as it encountered its limit-point condition in that realm of the ‘pure multiple’—or that formally unrestricted set-theoretical domain—which required that statements of the relevant class be open to the test of self-reflexive application.
In effect, the very confidence initially displayed by Frege and Russell in the power of their logical language (technically speaking: that of the first-order quantified predicate calculus) to offer a complete and perfectly consistent formalization of the set-theoretical domain was itself a sure sign of the project’s being headed for just such an obstacle somewhere along the way. It is precisely this recurrent gesture of containment—this move to control and delimit the scope of enquiry through various techniques of always premature ‘totalization’—that Badiou regards as having posed a chief obstacle to progress by evading the radical or, indeed, socio-political order. Hence his objection not only to the claim that intuition might yield valid insights or conceptual progress (since intuition is most often just the name applied to preconceived belief) but also to that narrowly logicist idea that the exploratory scope of set theory might be circumscribed by a purely formal programme whose terms are specifically in advance and which therefore pre-emptively restricts any future developments to what is conceivable at present. So it was, Badiou thinks, that the Frege-Russell project was predestined to run aground on those paradoxes that sprang to view as soon as it encountered its limit-point condition in that realm of the ‘pure multiple’—or that formally unrestricted set-theoretical domain—which required that statements of the relevant class be open to the test of self-reflexive application.
I should offer some further detail at this stage since the
episode in question is among the most crucial for Badiou’s understanding of set
theory and of the complex relationship between genesis and structure that has
characterized its history to date. The great promise of set-theory as envisaged
by Cantor, Frege, Russell and its other early proponents was that of reducing
mathematics to a purely logical or axiomatic-deductive structure of entailment
relations that would leave no room for anomaly or paradox. That claim
encountered its first major setback when Russell showed—by purely logical means—that
set theory was intrinsically prone to generate just such problem, namely the
kinds of self-reflexive, self-predicative or auto-referential paradox that
resulted from its dealing with formulas such as ‘the set of all sets that are
not members of themselves’ or ‘he who shaves the barber in a town where the
barber shaves every man except those who shave themselves’. Yet, as Badiou
points out, despite their somewhat contrived appearance such paradoxes all
derive from a basic formula (that of the set which is not a member of itself)
which, so far from being forced or extraordinary, in fact turns up—and quite
acceptably so—in each and every possible specification of a set. Thus ‘it is
obvious that the set of whole numbers is not itself a whole number’, and so on
for any range of similar instances (p. 40). To this extent it is an inbuilt
feature of set-theoretical thought, one that arises whenever it is a question
of asserting ‘the constitutive power of language over being-multiple’, and
which therefore cannot be regarded as something pathological or (as Russell and
Frege supposed) in need of surgical excision. However, it does take on such a
negative, subversive or system-threatening aspect when its implications are
followed through in the context of an ultra-logical programme which identifies
truth with formal validity, in turn, with the classical ideals of consistency
and total closure under logical entailment. In that context the acceptable face
of self-reference—its ubiquitous and therefore unobjectionable presence—undergoes
a distinct change of expression and becomes, in effect, the un-doer of that
whole optimistic logicist project.
Russell’s answer was to make it a stipulative rule that
statements in formal languages such as those of mathematics or the logical
calculus should not be self-referring in a way that gave rise to difficulties
of this sort. Rather they could best be averted by a ‘Theory or Types’ which
distinguished clearly between various orders or levels of statement, that is,
those belonging to the first-order language of direct or mater-mode assertion,
those that referred to such first-order statements from a higher logical level,
and so on up through successive stage of increasingly abstract formal (i.e.
meta-linguistic) specification. Only thus, Russell thought, could set theory—as
a crucial component of present-day developments in logic and mathematics—be kept
on its path towards an ever more secure, since ever more precisely codified
conception of validity or truth. Still his purported ‘solution’ to these
problems struck many, then and now, as objectionably ad hoc and as having more
to do with interest of pragmatic or methodological convenience than with
principles self-evident to reason. Indeed, the set-theoretical paradoxes have
remained a spur to philosophic thought and a potent source of speculative ideas
both within mathematics and across a range of other disciplines ever since
Russell first discovered them. Their impact was intensified by various related
developments, including—most notably—Gödel’s undecidability-proof to the
effect that any formal system sufficiently complex to generate the axioms of
elementary arithmetic or first-order logic would necessarily include or entail
at least one statement the truth or validity of which could not be proved within
the system itself. In other words, one could have either truth as matter of
rigorous logical procedure or consistency (‘completeness’) as a matter of
intra-systemic coherence but surely not both unless by some maneuver, like
Russell’s, that looked suspiciously like a mere device for saving
logico-mathematical appearances. Nevertheless set theory survived these and
other challenges through the effort of various thinkers to provide some method
of formal restatement in axiomatic terms that would keep the paradoxes safely
out of view or at least prevent them from doing real harm. During the past
century it has become absolutely central to every branch of pure and applied
mathematics, as well as to every mathematics-based development in the physical and
even (in certain contexts) the social and human sciences.
-Christopher Norris, Badiou’s Being and Event, pgs. 52-55.
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