NOTES
FROM: Badiou, A (2011) ‘The ontology of
multiplicity: the singleton of the void’, EGS
video.

[An
astonishing combination of mathematical and
political critique.Classic
example of philosophical arguments which veer
from one to the other]

If we
begin with nothingness or the void, we can only
begin to philosophize about reality by using a
number of definite simple operations.The
preferred one is to say that the void is in a
special set, the void set.This
set has only one element in it, so it is a
singleton set.In fact, by performing this operation
we’ve already established a difference between
nothing and one.It so happens that the difference between
zero and one is very important in our digital
world.The
basic difference between zero and one is the
origin of all differences, it is pure
difference, the paradigm of all differences, as
we see with the concrete example of the
development of binary code.

Succession
is another operation which can be seen in terms
of set theory as putting all the elements of X
into a set, and adding X itself.Again
what we’re doing here is putting elements in and
then naming the set of elements.We are
adding nothing extra, only something and its
name.Returning
to the void, we can see that the void has a
successor in the singleton set that we have
defined above.However, this added name must not itself
be an element of X: the name of something is
always external to the elements we have added,
the name comes from the exterior.As a
homely example, we can think of our own proper
names which come from outside, and clearly
reflect some arbitrary nomination, something not
integral to us.In this sense, we are all successors
because we’ve all been named externally.

There is a
problem with applying set theory to humans,
which is seen if we try to equate human beings
with their name.This is a form of reductionism, as when
our identity documents or unique identifying
number are taken for us our selves.Anyone
can see this is a dubious reduction, even though
it is perfectly possible.All
humans have qualities which are outside their
names.

Successions
make up a series.There is the void, the singleton set ,
the successor of the singleton set, the
successor of that successor and so on.We can
see this in the series of numbers one, two,
three and so on.We have an operation of pure repetition,
one that creates new terms.And we
began with nothingness and pure difference,
using only very simple operations.

These
processes of repetition and difference are very
important in our world, since they underpin for
example the circulation of money.Money
is already a numerical form, price is a relation
between the thing and an abstract number, and
the system of numbers itself.The
system of numbers, with its system of repetition
and difference, underpins the entire world.This
is also a reduction.We
know that crises can sometimes arise when the
money system itself proves to be inadequate, and
a lot of concrete crises actually reflect back
on the abstract nature of numbers.

Price is
always finite, and never infinite.If in
our world everything has a price, everything is
therefore finite, everything can be bought, even
the human subject as a price.What
we are describing here is a general definition
of corruption.If we see corruption is necessary and
inevitable, that arises from the global
projection of the market is a universal form.It is
an affirmation of the radical finitude of the
world.

Classical
philosophy also affirmed the finitude of human
life, but that led them to think of and desire
the infinite. This
is what Plato called the Ideal for example. Even in
their days, this was seen as disastrous, as
hubris, and this stands lies behind the
contemporary struggle against great ideas.These
are seen as pointless, a desire for the infinite
which is absurd because it has no price.Those
who desire the infinite can also not be
corrupted.

So there
are very concrete implications here.Our
world is necessarily corrupt, and this arises
from the projection of the scheme of succession
and difference [in quantification generally as
well as price?].Is anything without price?We
commonly referred to priceless things, and these
must be outside of the finite world.Perhaps
the purpose of philosophy is to create new
subjective and intellectual means to find the
priceless, if indeed that is possible.This
is not going to be easy because by definition it
must be something beyond our world, something
infinite. A new vision of the world is required.We
have seen lots of examples of artistic creation
or scientific innovation which have been rapidly
reduced and brought back within the system of
prices [so we need a proper philosophical
stance].

We’re
talking here about a requirement to change the
entire world, not just to find personal freedom.The
world affirms the law of numbers and that
everything is finite, so how do we get to the
infinite?If
we would affirm that the finite world is false,
we must be priceless ourselves!It is
necessary to go beyond the finite and how it
exists, to construct and rethink the infinite,
but not as just an ideal, but as a necessary
part of the world.

We search
for the infinite using set theory.We
search for the omega at the end of the finite
series, we need to bring into existence the
priceless, something that cannot be reducible to
pure difference and succession as above.We
need to avoid corruption in this broad sense,
the idea that everything has its price, the idea
that produced the money system.The
first stage is to firmly reject this view.Of
course we have to be in the world and cannot
escape it, but there are different modes of
living [in effect a simple or tactical
acceptance of corruption].To
refuse to adopt this critical will is itself the
beginning of corruption, a form of ‘passive
corruption’.We can be passively corrupted in this
sense even if we remain poor.

We need to
search for omega.It would be at the opposite end of a
series from the singleton, it will be the set of
all finite numbers beginning with the void, and
it will add a name from the outside.This
will move it beyond pure difference and outside
normal finite succession.However,
this omega can be seen as in a succession
itself.We
can certainly think of one using the procedure
and definition above (including all the
elements, the name omega, and another name from
the outside, say omega2).So a
first definition of the infinite, omega, is a
point between two successions, a finite one and
an infinite one.As a gateway to infinite succession, this
initial discovery will lead to a series of whole
new worlds appearing in the infinite (there is
no return to the finite here).Since
omega is a point beyond finite repetition, it
must be the start of a new succession, and at
the end of that sequence, a new infinite point
emerges, so we get a series of infinite
possibilities.

This can
be concretized by thinking about particular
artistic innovations that changed the world of
art and opened endless new possibilities.These
are genuine escapes, not just repetitions, and
therefore beyond corruption.So
there is a rational basis for these
possibilities.We have combined mathematical reasoning
with the other ideas here, and used mathematics
to provide new means of understanding.We
have actually discovered differences between the
infinites, a difference inside the infinite.The
infinite is not just an abstract possibility,
but has a positive existence, and there is a
multiplicity of different infinites.

It is
difficult to think of this sort of multiplicity
using normal intuition and we need abstract
definitions first, again using set theory.This
will help us clarify the notion of the infinite.The
whole argument is an example of how
mathematicians produced new and clear ideas
which have subsequently informed philosophy.