Alain Badiou: From Nothing to
Infinity
Towards the end of the third
lesson (a central point), Professor Alain Badiou named the void with Ø. Ø is
the name of nothingness. It is at this very point that he remarked on an
apparently marginal fact: Ø is a Danish letter. The argument that then followed,
constructed along a rigid mathematical procedure, led us to two different kinds
of infinity.
I will briefly reconstruct it
for my attentive reader, who will be eventually remunerated for his patience.
Ø names Ø, in other terms: /Ø/
names Ø – in order to distinguish the name from the object. A name is a set of
things – even if they are no/things. At this point we see (not without
surprise) that we have already an element – starting from nothing. We are going
to call this first element B. B = /Ø/. At the same time we are mathematically
obliged to notice that we not only confronting a first element, but that we
even have two elements: Ø and /Ø/. In other, mathematical, terms, we now have:
/Ø, /Ø//. We can easily acknowledge that /Ø, /Ø// is a new, third, element –
which we call BA. BA = /Ø, /Ø// or BA = /Ø, B/. This procedure can be logically
pursued ad infinitum.
On the 5th August 2011, Badiou illustrated
on the white-board the first steps of this progression from the beginning of
nothing to infinity – steps that I´ll repeat here once more.
On the left side the reader
finds the short version, while the respective extended version lies on the
right side. The names are arbitrary, contingent, or even necessary – as it will
be finally demonstrated – here.
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Ø
|
Ø
|
|
B = /Ø/
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B = /Ø/
|
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BA = /Ø, B/
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BA = /Ø, /Ø//
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BAD =/Ø, B, BA/
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BAD = /Ø, /Ø/, /Ø, /Ø///
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The reader might experience the
sense of the process here, while I will nonetheless extend and re-propose it
for a few further passages:
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BADI = /Ø, B, BA, BAD/
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BADI = /Ø, /Ø/, /Ø, /Ø//, /Ø,
/Ø/, /Ø, /Ø////
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We see that names, like numbers, are made
of nothingness. But this process of nothingness proceeds towards infinity.
Certainly this is the negative infinity of the apeiron, without-limit. We experience the nothingness of
Succession: the novelty of nothingness in endless repetition of difference.
Infinity is then the negative necessity of the finite, which enters the finite
as its inner limit and strength. But two steps more will help us go to the root
of the argument between infinity and nothingness:
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BADIO
= /Ø, B, BA, BAD, BADI/
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BADIO = /Ø, /Ø/, /Ø, /Ø//, /Ø, /Ø/, /Ø, /Ø///,
/Ø,/Ø/, /Ø, /Ø//, /Ø, /Ø/, /Ø, /Ø/////
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From this we deduce our last step of the
succession – which – keep in mind – proceeds ad infinitum:
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BADIOU
= /Ø, B, BA, BAD, BADI, BADIOU/
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BADIOU = /Ø, /Ø/, /Ø, /Ø//, /Ø, /Ø/, /Ø, /Ø///,
/Ø,/Ø/, /Ø, /Ø//, /Ø, /Ø/, /Ø, /Ø////,/Ø, /Ø/, /Ø, /Ø//, /Ø, /Ø/, /Ø, /Ø///,
/Ø,/Ø/, /Ø, /Ø//, /Ø, /Ø/, /Ø, /Ø//////
|
I ask my attentive reader to focus on this
extended equation for a few seconds and “experience” nothingness ad infinitum.
My reader should not underestimate its
value, considering that history and culture, language and cakes, pop-corn and
Gods, anything else imaginable (and more importantly: not imaginable) could
enter the equation as a new name of nothing. And yet we must accomplish a step
more if we want to grasp infinity not merely as a negative concept. In fact, if
infinity is the infinity of repetition, it´s still inside repetition. We must
then think of an infinity outside all names, something outside Succession. This
is enough to suggest that succession (of names) constitutes a unity external to
the principle of succession, since this unity cannot succeed in the succession,
therefore belonging as such to a different, totally other register than
succession. The space of this total otherness would be the positive definition
of infinity which, mathematically, could be read so: ¬Ǝx[Ѡ=S(x)], that is: there is (Ǝ)
no (¬) x for which infinity (Ѡ) is the successive (S) number. In this case,
infinity would constitute the weakness of finitude – since it would be the
possibility of something outside finitude and repetition. Mathematically we can
distinguish infinity as the space of repetition (Ѡ) and infinity as the without
limit of repetition (ad infinitum):
Ѡ =
/B, BA, BAD, BADI, BADIO, BADIOU…, ad infinitum/
It is
at this moment that Badiou generously showed the relevance of this
mathematization to every-day life, asking us: “What´s a woman?”
He
suggested that we should not think of “woman” as the infinite negative limit
rather as the infinite other beyond finitude, the weakness of finitude. It is
at this point that I remembered that first precious reminder: “Ø is a Danish
letter” and said to myself: What kind of letter is Ѡ?
Ѡ =
/Ø, BADIOU, Ø/.
I must
say that Badiou´s algebraic handwriting unequivocally helped me to distinguish
there, in that Ѡ-shaped letter, a generous breast – and I let my reader draw
his own conclusions out of this purely, exclusively mathematical deduction of
infinity from nothingness. It´s the miracle of pure mathematics!
P.S.
Additionally,
following a strictly mathematical deduction – one can also extrapolate
from ¬Ǝx[Ѡ=S(x)] my personal reception
of Badiou´s main thesis on infinity. To the aim it suffices to think that E is
the first letter of my name, which, as a sort of oppositional convulsion (in
listening to Badiou´s lesson), gets turned with contrariety, in a contrary
direction.
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