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.

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