There are two philosophical approaches that can be compared: the one of Deleuze and the one of Badiou. In this post I want to sketch my superficial understanding of their ideas and especially concentrate on the role mathematics plays.

[With superficial I mean that I read a lot and grasped something, but also have a lot of questionmarks. I try to delineate what my understanding of their approaches is at the moment. I sometimes hint at passages and secondary literature that lead me to this view, but it is not something based on close reading. I want to come back to this sketch in later posts to prove, improve or disprove it with primary text passages. This post has therefore transitional status and I don’t guarantee its correctness.]

## 0. Overview

## 0.1 Deleuze

Deleuze’s philosophy can be summarized in the formula: Monism = Pluralism. What does this mean? The monism part can be seen as another description of immanentism. To have an immanent philosophy means that you have one world, everything is inside this world and there is no other realm (like for example Plato’s heaven of ideas or Kant’s transcendental conditions if they are understood as different realm outside reality) that structures reality.

The pluralism can be seen as an anti-essentialist, pluralist methodology. [This part is inspired by DeLanda and other philosophers who refer to Deleuze, but – especially the word I use – are not based directly on Deleuze’s own writing.] Essentialism is here understood in its minimal definition: A is B iff (if and only if) A has property C. C is therefore a necessary condition to be B. For example: This tree is only a tree iff it is wooden. Necessary conditions can be understood as essential properties of something. Aristotle for example can be read in that way. For Aristotle there are necessary and accidental properties. The necessary properties are connected to the essence and the accidental properties generate the differences between certain instances of a group of entities with the same essence.

[Short note on Aristotle: There is a different way to read Aristotle that is based on his theory of science and language. Aristotle talks a lot about the way we ask questions. Therefore you can read Aristotle’s necessity (or at least most notions of it) as question dependent. In his *Physics* he talks about a dugout canoe and asks the question: Is it natural or artificial? Then he goes on to refine the question. If we want to know how this object was created, it is artificial. People used tools to bring the wood in this form. If we want to know how this object behaves in the future (lying around in the dirt), it is natural. It is prone to rot and wither away. This gives rise to the reading that Aristotle’s aitiai and archai (causes and origins/principles) are question- and language dependent.]

Deleuze is non- or anti-essentialist because there are no necessary properties. On the one side Deleuze is not so much interested in products or results, but in the processes that generate them (and can change and destroy them). The talk about necessary properties is often static and without history.

On the other side Deleuze wants to allow a multiplicity of functions and perspectives. One of the famous concepts of Deleuze is the body without organs (short: bwo). The word organ stems from the Greek word for tool (organon). A tool has a function and an organ can be understood as a function of a body. In *Mille Plateaux *the chapter about bwo starts with different examples of bodies losing their organ. One of it is a passage from Burrough’s *Naked Lunch*. This passage imagines a (human or at least animal) body with only one orifice: to eat, to shit, to fuck. In the first episode of the Deleuze podcast *buddies without organs* (https://buddieswithout.org/) this example is extended by a christian arguing against homosexuality. The anus can be considered an organ. But what is the function of the anus? Only to shit? Is anal sex wrong because we disregard the right function of the organ? For Deleuze that is clearly not the case. Where does this function come from? What makes a certain function the right function?

At least one way to understand the bwo is to say that a body has no proper function. [There is more to the concept of the bwo, but I think this is the important one here.]

Transcendent structures can be critiqued in two ways: They are not only adding another realm to the world, but they structuring reality (via the transcendent realm) in a certain way that delivers correct properties and functions.

The bwo can be seen as methodological point to look at the world without inscribing functions. The near impossibility of this project can be felt in the passages describing it. To be aware of the functions we inscribe is nontheless an achievement of awareness.

Let’s summarize so far: For Deleuze the world is one realm to which we should add nothing external. The world is structured by processes and forces. We cannot give this processes a certain function that determines them.

The central questions for Deleuze’s project are then: If we don’t look at results, but at the processes, we recognize becoming. We realize things change. How can we describe changes without functions? Is novelty a possibility in an immanent world? How can the new be generated? Does a novelty add something to the world or is it a change in the same world (and what exactly changes)?

## 0.2 Badiou

As well as for Deleuze there is a formula that expresses the main idea of Badiou’s philosophy: Mathematics = Ontology. Ontology is concerned with being. But ontology as well as mathematics isn’t concerned with beings, it is concerned with being as being. Let’s unpack this strange sounding words a little bit. Mathematics is abstract. Badiou is concerned with a certain strain of mathematics: set theory. Set theory is concerned with sets and their elements. A set comprises elements under a certain definition. Unlike Deleuze, Badiou is concerned with truth as a notion of universality. For Badiou the world is multiple (very simple: full of different things). A set as a truth establishes a category. It’s universality means that it is true in all times and can therefore reoccur.

One prominent representative and (for many) the founder of set theory is Cantor. Cantor was not only concerned with sets, but also with infinity. He realized that you can construct mathematically larger and larger sets of infinity. [The following is very much simplified and misuses some terms for a simpler understanding:] Let’s call the infinity of numbers a continuum. Can we construct an infinitely large number or notion of infinity that comprises all numbers? Cantor found out: No, there is no infinity, to which no larger infinity can be constructed.

In Badiou’s work this is the foundation. When truths are discovered or expressed we discover a certain aspect. But as Cantor shows, there are an infinity of truths and we cannnot even express this infinity. There is always something more. This more or surplus means for Badiou that something can happen – an event – that shows and expresses something that wasn’t realized before (more detailled: for Badiou the event creates something indiscernible, that cannot be discerned by privious knowledge and thereby demanding a new knowledge and truth). It forms something new – a novelty – that disturbs all our privous knowledge. For Badiou we have to be faithfull to such events and discover their truths. We must bring them in the language of universality. (For Badiou this counts for scientific discoveries as well as political events. His emancipatory politics is looking for the excluded. The excluded – e.g. immigrants – express their situation and (a new) universality is demanded that includes and respects them.)

A summary so far: Badiou considers himself a materialist. But the material world is infinite and this infinity can always be grasped partially and never completely. Events reveal new truths, to develop them fully we must stay faithful to them, demand their universality and rearrange our previous knowledge.

[A short comment: Quentin Meillassoux – a disciple of Badiou – uses Badiou’s set theoretical argument to show that the future will always bring new events that are unforeseeable. I’m not sure if Badiou himself sees infinity as a property of the world or also as a property of time. At the moment I can only see an infinite world in Badiou, but not a time that is able to generate the creatio ex nihilo like Meillassoux does.]

## 0.3 Similarities

Now we can compare these two approaches: As in every comparison you can name similarities and differences. Let’s start with the similarities.

Both use current science and mathematics to understand how new methodologies introduced in the 20^{th} century allow us to look at the world in another way. One important question for them is novelty: If the world can change and there is becoming, how can we describe the upcoming new? They both try to use their results to better understand capitalism and how to overcome it.

## 1. Mathematics

## 1.1 Deleuze

Let’s look at the differences. First the role of mathematics: Expecially when read with DeLanda – Deleuze’s use of mathematics is primarily the mathematics of complexity as it is introduced in the theory of dynamic systems. In dynamic systems you look at the world, select a system you want to describe, decide which parameters are relevant and than you can construct a dynamic system. These dynamic systems allow us to look at the behavior of them through time. Via some mathematical techniques it is possible to construct vectors and vectorspaces. These vectors show us tendencies of the systems (singularities, attractors).

Deleuze interprets these techniques ontologically and offers a view where a new modality arises: virtuality. The virtual is real but not actual. Meaning attractors and singularities are not actualized (e.g. already happened), but structure the behavior of systems and are therefore nontheless real.

This view of mathematics can be called immanent, because we use mathematics to describe what is happening in the world. These are real tendencies discovered by mathematics. This leads also to a view of processes. We don’t look at entities as results or structure the world via logic. Morphogenetic processes matter and explain the results (and how even the results can change again).

## 1.2 Badiou

Badiou on the other side goes into abstraction. His mathematics are not dynamic systems, but set theory. As far as I understand him he has the equation: mathematics = ontology. Mathematics and ontology are the same because they explain truth procedures and abstract themselves from concrete entities. They are concerned with universal truths. This is nothing that can be completed. Mathematics – and therefore ontology – have a history. It is impossible to anticipate the changes in mathematics and the resulting consequences. Therefore ontology can change and new universals appear. Philosophy is concerned with events. An event is a historical and can propose a new concrete universal. In *The Communist Hypothesis* he explains that communism is such a universal claim. It is an event that is aimed at a universal goal. Because this goal wasn’t reached, but nothing can tell us that it cannot be reached, it is a hypothesis.

The important point is that mathematics is not something in the immanent realm of the real, but connected to thought and formal procedures that allow us to grasp the real.

## 2. Novelty

## 2.1 Deleuze

In *Logique du Sens *Deleuze introduces two versions of time: chronos and aion. Chronos is the time of a strict materialsm. Chronos is the time that can grasp bodies and their relations. It can grasp cause and effect in the sense most natural scientists talk about it. Aion on the other side describes uncorporeal effect on the surface. Aion can be identified with the virtual (see for example Meillassoux’s contribution to *Collapse III). *The processes of the virtual allow to change the nature of chronos – the materialist, determinist conception of reality.

## 2.2 Badiou

Badiou explains novelty on another level. As explained above, ontology is never complete. Therefore we cannot have a complete understanding and novelty will always be possible.

His disciple Quentin Meillassoux goes even further. He shows the problem of probability. To have a probability you need a set of possibilities. But how to construct these possibilities. If we never have a full knowledge about all possibilities, we can never make a full description and therefore never have a real probability. Novelty is ex nihilo, not in the sense that it has no cause, but in the sense that there is more in effect than in the cause – something that is unforeseeable.

## 3. Truth

## 3.1 Deleuze

Deleuze is no philosopher of truth. His approach is more concerned what can be called problematic or relevant. For Deleuze there are problems which must be expressed in concepts to find actions to react to them. In his Deleuze interpretation in *Intensive Science and Virtual Philosophy* DeLanda gives an account about how this can be understood as an approach to the theory of science. The problematic approach is not concerned with truth, but with detecting relevant parameters. I’m not completely sure, if that is really all that is to Deleuze. Especially with regard to his Nietzsche interpretation there seems to be something more.

The important point is Deleuze doesn’t seek truths – truths are not part of his philosophy. He is involved in processes and problems that need understanding, but not definite or universal truths.

## 3.2 Badiou

Badiou on the other side wants to restore the notion of truth – as should be obvious from what I have written above. Truth is very important in his philosophy. A truth is expressed as universal. It remains universal. New truths don’t change that universality, but require us to rethink the arrangement of previous found truths.

## Problems

Here I want to hint at problems with my above superficial exposition of Deleuze. It’s not to explain anything above, but to hint at problems in my own sketch.

One of the difficulties of Deleuze is his relation to truth. As well as Nietzsche he is highly suspicious of truth. In *Intensive Science and Virtual Philosophy *DeLanda tackles this problem and offers a reading of a problematic approach. The criterion is a well-defined problem. It doesn’t matter if you have all the real parameters, but it matters to have a problem that can tell you something in regard to a question and in regard to a possible solution.

But that doesn’t resolve the problem of truth. If you decide what is important or relevant you need to have criteria again. And talking about the real, presupposes a knowledge about the real. One solution could be that truth is a discursive term, caught in its own set of logic. Then you can say that talking about the real is not in terms of truth, but in terms of adequacy and adequacy depends on the questions and terms you start with. [And you end with something similar to the Aristotle interpretation I hinted at in square brackets in 0.1.]

Another problem is: What is mathematics for Deleuze? The chapter about the smooth and striated in *Mille Plateux* discusses the different uses of numbers: to count, as unit of meassure, etc. So there is not one fixed meaning of numbers. A possible approach is to see mathematics – in its abstraction – as something that is possible to highlight different aspects of the real (the actual as well as the virtual). Each use of mathematics bound to a different problem.

At least one problem still remains: If mathematics is immanent and not simply a representation, what is it? How can we guarantee to grasp the real inside mathematics?

This connects to similar problem: How can we distinguish the immanent from the transcendent? For me it is not always clear that a transcendent realm is really an unnecessary addition to the immanent realm. Most transcendent conceptualisations are gained via analysis of things or processes. How is this exactly different from Deleuze’s view on mathematics?

## Conclusion

In this sketch I used Deleuze and Badiou as two different approaches to contemporary problems. They both involve contemporary mathematics to grasp our world in a way that wasn’t possibly before the arrrival of new forms of mathematics. As I hinted at in the problems there is a lot of work to do to get a better grasp of these approaches. In my opinion these approaches are important because they offer us a new way to look at reality, possibilities (or better: modalities, because the virtual is not the possible), science and how all of this is connected to society (planning, risk-management and economics are only a few areas where models have a huge impact on our daily lives – even if we don’t always realize it). To understand models and what they imply is important. In the future I want to explore these problems in more detail, but I hope this sketch will help others to get an overview about what I’m talking about and what is at stake.