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In his book Logics of Worlds, Badiou gives a formal mathematical definition of a “world”, but does not construct a single concrete example. In this note, I give a concrete example in which the objects of the world are materially-embedded stochastic processes (i.e., stateful processes unfolding over time in the universe).

This construction has the property that one does not have to specify the transcendental or identity values explicitly, but instead can actually calculate the values starting from an explicit definition of an object. Moreover, its multiplicities are transparently the sort of things we encounter in the world: material processes evolving over time according to some logic.

As I understand it, one of the core aims of Badiou’s project is to show that the world that we live in does not have to be this way — that nothing about the social order is fixed — and to point towards how we ought to orient ourselves in order to aid the development of a new world to replace this one.

That conclusion in a narrow sense is a perfectly obvious if trivial: it follows from the second law of thermodynamics. Nothing is eternal. What is more interesting is to develop a close analysis of why a thing is not eternal. A given system, say a society $S$ is embedded in certain logics which are not its own. For example, it is physically embodied and that’s why we can be certain it will perish due to the accumulation of physical entropy if for no other reason. However, we can get much tighter bounds on its lifespan by examining its embedding into and containment of more abstract logics, e.g., the logic of capital which demands the destabilization of the biosphere on a very short timeline.

I suppose what Badiou’s project is, from this perspective, is to show that the logic of a very bare conceptual apparatus (that presumably any human cognition would necessitate) already implies the impossibility of a fixed social order. However, arguing from such a high level of abstraction will yield uselessly loose bounds on any given social order. An actual understanding of the embedding of a given social order in the social and physical world is necessary for yielding more actionable information.

Part of his project too is to distinguish being from appearance. To separate the real meat of what something is apart from how it concretely manifests in the world. This I think he does well but the models must be brought closer to reality to see how the rubber meets the road in his formalism.

Despite my critiques, I think his attempt to develop precise philosophical categories for political purposes, described in mathematical language, is very valuable. The construction of communism is a phase shift for humanity. It is something that will take a long time and a very deep transformation in ourselves. Given the enormity and long-term nature of the task, I think it is well worth it to attempt the slow task of building up a mathematically precise conceptual framework to ground our action. Not as the sum total of political action of course.

The central object of Badiou’s book is a “world”. A world is a mathematical object which includes

• “Multiples” – the kind of thing (an ideal type) that may exist in a world in different concrete instantiations. An example that he refers to several times is that of an “anarchist”. If there is indeed such thing as an “anarchist”, we ought to be able to list all the concrete instances of people in the world who are being, to some degree, an “anarchist”.
• A “transcendental” – a structure which measures how much any given concrete instance of a multiple “exists” in the world. In our example of an anarchist, it would be used to measure how much some particular person is being an anarchist.

A world consists of (p 339)

• A set $\mathbf{m}$ of sets, closed under taking powersets. The elements of $A$ are called “multiples”. The elements of a multiple $A$, Badiou calls “real elements”, I will also call them “concrete instances” or “instances” of the multiple.
• A frame (or equivalently, complete Heyting algebra) $T$. That is, $T$ is a poset with a minimum $\mu$, where any set of elements has a join (least upper bound), and any two elements $p,q$ have a meet $p \land q$ (greatest lower bound). This will give us a space in which to measure the degree of existence of beings. The maximum element of $T$ will be called $M$.
• For $A \in \mathbf{M}$, a function $\mathbf{Id}_A \colon A \times A \to T$ which measures the degree of identity of any two instances of a multiple. When it’s clear from context, I will omit the subscript $A$, as he does. This function obeys a few axioms. Namely, for any $x, y, z \in A$, $\mathbf{Id}(x, y) \land \mathbf{Id}(y, z) \leq \mathbf{Id}(x, z)$ and $\mathbf{Id}(x, y) = \mathbf{Id}(y, x)$.¹
• For $x \in A$, $\mathbf{E}x := \mathbf{Id}(x, x)$ is called the “existence of $x$” and is supposed to measure how much $x$ exists in the world $\mathbf{m}$.[^existence]

Situating this definition mathematically

Let’s pause here to consider the definition. From a logical point of view, it can be thought of as something like a type theory with an equality type. The transcendental $T$ would just be the sort that the equality type is taking values in, for example a type $\mathsf{Prop}$ of propositions. In that case, $\mathsf{Prop}$ is ordered by implication (i.e., $P \leq Q$ iff there is a term of type $P \to Q$), with the meet $\land$ given by conjunction/product.

From a topos-theoretic point of view, we are dealing with something like a Grothendieck topos. I.e., the topos of sheaves over the transcendental $T$, $\mathbf{Sh}(T)$. A multiplicity $A$ would correspond to a sheaf $\rho_A$ with $\rho_A(p) = \Set{x \in A \mid p \leq \mathbf{E}x}$. We can restrict appropriately since if we have $p \leq q$ and the existence of $x$ is at least $q$, then it is at least $p$ as well. How to interpret $\mathbf{Id}$ generally is a little more complicated.

Towards a model

Now, such a thing is supposed to represent a “world”, where the multiplicities are the objects of ordinary experience (chairs, societies, anarchists, etc.). Let’s move towards that concept by defining a world.

We’ll start with the general idea.

First, fix a homogeneous stochastic process $\mathcal{U}$ on a state space $U$. A stochastic process is a time-indexed family of random variables. Let’s say time is a partially ordered set $\mathsf{Time}$. To take a concrete example, think that $\mathcal{U}$ is Conway’s Game of Life on the plane with time equal to the natural numbers $\mathbb{N}$ and initial state chosen at random (according to some reasonable Borel measure on the set of configurations $\Set{0, 1}^{\mathbb{Z} \times \mathbb{Z}}$).

We want to be able to talk about entities within this world. For example, abstract entities like “an exponentially growing configuration”, “the execution of the Turing Machine $M$”, or concrete entities like “a glider”. Such entities are things that have their own internal logic. These examples are all deterministic, and so we think of them as mere dynamical systems, which can be associated to a stochastic process by picking the initial state at random. On the other hand, we can also simulate many genuine stochastic processes within the game of life world, by stuffing randomness somewhere in the world, where it can be retrieved when necessary.

The idea is that all of these processes can be simulated within the game of life. Therefore, in any configuration, or run of the game of life, we can look at it and decompose it into simpler processes at many levels of abstraction: e.g., we may look at a run and say, “this consists of the execution of the Turing machine $M$” or we may look at the same run and say, “this represents 10,000 gliders”. This is just as when we look at a concrete biological organism, we can interpret it as a bunch of physical particles interacting, as specified micro-biological mechanisms unfolding (the transcription of proteins, etc.), as specified macro-biological processes unfolding (the digestion of food, the propagation of an action potential), as the high level logic of an organism itself.

A multiplicity will thus be another stochastic process. This represents an ideal concept that may have some material manifestation. The “elements” of this multiplicity will be purported material manifestations, and the existence degree of each element will measure the degree to which eeit really is a material manifestation of the process.

Let’s formalize this.[^formalization]

• A multiplicity will be a stochastic process $\mathcal{A}$ with underlying state set $A$.
• In a divergence from Badiou, a multiplicity is not itself a set, but we associate to it a set. The set we associate it to is the set of partial functions $\mathbf{L}_\mathcal{A} := \Set{f \colon U \rightharpoonup A}$. In other words, each such $f$ is a lens on the universe, associating to a given state of the universe the state of $A$ which it is interpreted as encoding. We allow this lens $f$ to be defined only on a subset rather than all of $U$ so that we can avoid having to deal with configurations of the universe in which our intended instance does not even appear, in which case we’d have to output some phony state of $A$. For example, if $A$ is the execution state of a Turing machine, we can’t recognize what state the game of life board is representing if it doesn’t represent a Turing machine at all.
• Badiou demanded the set of multiplicities be closed under powerset. I don’t think that’s really needed, probably better is that it should be a topos of multiplicities, which we do have in this case.
• The transcendental $T$ will be subsets of $\mathcal{P}(\mathsf{Time}) \times \Omega$. Elements $(S, \omega)$ of such a subset represent slices of the universe (these times with these initial conditions). Ordering is by inclusion, the minimal element is the empty set and the maximal element is the whole set $\mathcal{P}(\mathsf{Time}) \times \Omega$ (all times with any initial conditions).
• Define $$E(f) := \Set{(S, \omega) \mid \omega \in \Omega, S \subseteq \mathsf{Time}, \forall s \in S, f(\mathcal{U}_{s}(\omega)) = \mathcal{A}_i(\omega)}$$ In other words, $E(f) \in T$ is the set of runs of reality in which $\mathcal{A}$ is a good model for what is happening.
• Now define $\mathbf{Id}(f, g) = E(f) \cap E(g)$. It is immediate that this definition satisfies Badiou’s axioms. Symmetry holds since $$\mathbf{Id}(f, g) = E(f) \cap E(g) = E(g) \cap E(f) = \mathbf{Id}(g, f)$$. Transitivity holds since $$\mathbf{Id}(f, g) \cap \mathbf{Id}(g, h) = E(f) \cap E(g) \cap E(h) \subseteq E(f) \cap E(h) = \mathbf{Id}(f, h)$$

[]: I think this is far from the only formalization of this idea. It is merely the easiest one I could think of. Mostly likely there are ones that are better in terms of the ease with which existence degrees can be computed, and which are more conceptually valid.

Atoms in this model

In our model, Badiou definition of a “phenomenal component” of a multiple $\mathcal{A}$ is a function $\pi \colon \mathbf{L}_\mathcal{A} \to T$.

It’s basically a subobject of $\mathbf{L}_\mathcal{A}$ where truth values are measured in $T$.

An atom is a defined to be a component satisfying two axioms: $a(x) \cap \mathbf{Id}(x, y) \leq a(y)$ and $a(x) \cap a(y) \leq \mathbf{Id}(x, y)$.

These two axioms imply that for any $x, y$ which exist absolutely (i.e., their existence value is maximal), their identity value $\mathbf{Id}(x, y)$ is also maximal. Or in other words, there is essentially only one element which exists absolutely.

It is not clear to me what the atoms are in our model. Badiou’s postulate of materialism however does not hold. This postulate says that any atom $\pi$ is equal to $\mathbf{Id}(x,-)$ for some $x$. In other words, “the only atoms that exist are represented by elements of $A$.

Conjecture: It seems to me this postulate is not true in our mode for the following reason: $E$ is clearly an atom but if there is some $x$ for which $E(y) = \mathbf{Id}(x, y) = E(y) \cap E(x)$ for all $y$, then $x$ is an optimal model for $A$. I.e., it is correct whenever any model for $\mathcal{A}$ is correct.

Going forward

Going forward I would like to characterize the atoms in this model, and to understand better models of this sort, where the objects need not be stochastic processes, but some kind of state-machine like thing, which have a notion of embeddability in / simulation of each other.