**Vincent Ardourel / Brownian motion, infinite limit, and emergence of stochasticity**

Brownian motion is the random motion of a particle in a fluid due to its collisions with the molecules of the fluid. Its dynamics is generally described by a stochastic equation, viz. the Langevin equation, which represents the effect of the collisions. Yet each collision in the fluid is supposed to be deterministic since the molecules of the fluid can be represented by hard spheres. An important question for the foundations of statistical mechanics is whether the stochastic motion of the particle can be derived from a deterministic system of hard spheres. In this talk, I investigate this question from a recent mathematical derivation of the Brownian motion from a hard spheres gas (Bodineau, Gallagher and Saint-Raymond 2016). Its main interest is that Brownian motion is derived as the *limit* of a deterministic system. I thus clarify the role of the *infinite limit* in the appearance of stochasticity and I discuss whether this stochastic behavior can be viewed as an emergent behavior. In particular, I show that this discussion is closely related to lively debates on the role of the infinite limit in the emergence of behaviors within statistical mechanics, viz. phase transitions and irreversible behaviors.

**Ämin Baumeler / Reversible time travel with freedom of choice and its implications for computation**

General relativity predicts the existence of closed time-like curves, along which a material object could travel back in time and interact with its past self. The natural question is whether this possibility leads to inconsistencies: Could the object interact in such a way to prevent its own time travel? If this is the case, self-consistency should forbid certain initial conditions from ever happening, a possibility at odds with the local nature of dynamical laws. Here we consider the most general deterministic dynamics connecting classical degrees of freedom defined on a set of bounded space-time regions, requiring that it is compatible with arbitrary operations performed in the local regions. We find that any such dynamics can be realised through reversible interactions. We further find that consistency with local operations is compatible with non-trivial time travel: Three parties can interact in such a way to be all both in the future and in the past of each other, while being free to perform arbitrary local operations. This talk is rounded up by describing and analyzing a model of computation that exploits time travel; we show its computational tameness.

**Jean Bricmont / A strange sort of determinism**

The de Broglie-Bohm theory is a deterministic theory that explains what the measurements are in quantum mechanics (without introducing an « observer ») and that accounts for the apparent randomness of the quantum phenomena.

But the determinism of that theory has two special features:

- it is non local, so that to predict the future even for a small region of space one has to specify the initial conditions over a possibly very large region of space.
- those initial conditions are in principle uncontrollable beyond what is allowed by the wave function.

**Fabrice Correia / Indeterminism and the Growing Block Theory of Time**

According to the Growing Block Theory of time (GBT), the collection of all the things that there are constantly grows over time. In this, the theory is in contrast with its two (more) popular rivals, Eternalism and Presentism. I argue that there is a form of indeterminism which is compatible with GBT but not with Eternalism or Presentism, and that this fact is a point in favour of GBT.

(The talk will substantively draw on joint work with Sven Rosenkranz.)

**Juliusz Doboszewski / Making the notion of determinism precise in general relativity**

In general relativity structure of spacetime is dynamical and varies between solutions. This feature of the theory provides us with a dizzying variety of solutions which may be seen as indeterministic, but at the same time poses significant difficulties for a precise notion of determinism. In my talk I will present few candidates for a definition of determinism, and discuss required assumptions concerning global features of spacetime and the verdicts these definitions give in few interesting classes of examples.

**Nicolas Gisin / Non-determinism in Physics**

Quantum theory is a beautiful example of a non-deterministic physics theory. Admittedly, some interpretations supplement quantum theory with additional variables in order to make it deterministic, see e.g. Bohm’s theory that adds point particles and Many-Worlds that adds entire universes. However, to remain consistent these interpretations have to claim that these additional variables are de facto not accessible. Hence, quantum theory is, at least de facto, non-deterministic. The subtleties of various interpretations may only affect the character of this non-determinism, from truly fundamental and intrinsic pure chance to mere epistemic randomness.

I’ll argue that classical Newton mechanics is also non-deterministic. For this I start from the hypothesis that a finite volume of space can’t hold more than a finite amount of information. As a consequence, I argue that the mathematical real numbers are physically unreal. Indeed, almost all so-called real numbers contain an infinite amount of information, like, e.g., the answers to all questions one may formulate in any human language. Moreover, all real numbers, except a countable subset, are incomputable in the sense that their digits are random. Hence, a better name for them is random numbers. This name illustrates the fact that determinism can’t be based on the use of “real” numbers to represent initial conditions. Hence, Newtonian classical mechanics is not deterministic, contrary to standard claims and beliefs, except for stable systems like harmonic oscillators. However, the use of the mathematical real numbers is undoubtedly very useful as an idealization to allow for, e.g., differential equations. But one should not make the confusion of believing that this idealization implies that nature herself is deterministic: A deterministic theoretical model of physics doesn’t imply that nature is deterministic.

Consequently, in most physical dynamical systems, i.e. in chaotic systems, the initial conditions are random: after some determined initial digits, the next digits are undetermined (they don’t have any ontological existence). Pretty soon, these random digits drive the system. This raises the question as to when the undetermined digits get actualized, i.e. get determined. This is the classical analog of the well-known quantum measurement problem. I argue that such a problem arises in all non-deterministic models.

The unavoidable measurement problem (when do res-potentia turn into res extensa) motivate physicists to supplement non-deterministic theories with additional variables, with the hope of developing a deterministic theory with identical predictions. For classical Newton theory, the additional “hidden” variables are merely the real numbers. Note that these real numbers are de facto non-accessible, like Bohmian positions and parallel universes. I argue that one should not artificially make our theories deterministic, but accept the measurement problem as a real physics problem that will prove fruitful for new physics and philosophy, in particular for the quest of quantum gravity and for the debates on time and free-will.

**Anna Marmodoro / Probabilistics Powers?**

**Tim Palmer / Indeterminism vs Computationally Irreducible Determinism: New Approaches to Understanding the Bell Theorem**

This talk is in two halves. In the first half I will discuss a new approach to the Bell Theorem in which the metric of state space, conventionally Euclidean in both classical and quantum theory, is replaced by a p-adic-like metric. The physical motivation for introducing such a metric is the belief that the universe can be considered a deterministic dynamical system evolving on a fractal invariant set in its state space, and that the geometry of this fractal set is primal in determining the laws of physics at their deepest. The link to p-adic metrics arises because a Cantor set based on p iterated pieces is homeomorphic to the set of p-adic integers. Based on this framework, and on number theoretic properties of spherical triangles, the Bell inequality itself is shown to be undefined, and the Bell-like inequalities that have been shown to have been violated experimentally are p-adically distant from the Bell inequality. In summary, I claim that the Bell inequality has not been shown to be violated experimentally, even approximately – not so much a loophole as a gaping chasm in the interpretation of experimental results on the violation of the Bell inequality. In the second half of this talk I want to discuss whether there is a computational procedure (i.e. an algorithm run by some computational subset of the universe) which can determine whether two points in state space are close with respect to the proposed p-adic-like metric. The answer is no: properties of the fractal invariant set cannot be estimated by computationally simpler systems. Putting the two parts of the talk together, I claim that quantum physics can be understood in terms of deterministic locally causal dynamics – there is no need for indeterminism in fundamental physics. However, fundamentally, such determinism must be computationally irreducible.

**Renato Renner / Can quantum mechanics describe an agent who uses quantum mechanics?**

Quantum mechanics may be used to describe systems that contain agents who themselves employ quantum mechanics to make predictions. We propose a thought experiment to test the consistency of such a self-referential use of the theory. The experiment consists of an agent who, upon observing the outcome of a measurement, must conclude that another agent has predicted with certainty a different outcome for this measurement. The agents’ conclusions, although all derived within quantum mechanics, are thus inconsistent.