Lorenz Gödel and Penrose: new perspectives on determinism and unpredictability, from fundamental physics to the science of climate change

Lorenz is one of the pioneers of chaos theory. However, over 50 years before Lorenz, Poincaré discovered the sensitive dependence on initial conditions that characterises chaos. So what makes Lorenz’s contribution so important? I argue it is the discovery of the fractal invariant set in state space: the Lorenz attractor. Quite amazingly, properties of the Lorenz attractor can be shown to link the calculus of dynamical systems theory to deep and diverse areas of mathematics such as Wiles’ proof of Fermat’s Last Theorem and Gödel’s incompleteness theorem. But more than this, I argue that the fractal invariant set has implications for physics – not only for practical problems such as climate prediction, but also for the deepest problems of fundamental physics. In particular, I will put some meat on the bones of Penrose’s suggestion that “the correct theory of quantum gravity might be a deterministic but non-computable theory” by treating the universe as a dynamical system with fractal invariant set. The result is a novel perspective, not only on the quantum gravity programme, but also on quantum physics in general.