Incomputability in Physics and Biology

Computability has its origins in Logic within the framework formed along the original path laid down by the founding fathers of the modern foundational analysis for Mathematics (Frege and Hilbert). This theoretical itinerary, which was largely focused on Logic and Arithmetic, departed in principle from the renewed relations between Geometry and Physics occurring at the time. In particular, the key issue of physical measurement, as our only access to ‘reality’, played no part in its theoretical framework. This is in stark contrast to the position in Physics, where the role of measurement has been a core theoretical and epistemological issue since Poincaré, Planck and Einstein. Furthermore, measurement is intimately related to unpredictability, (in-)determinism and the relationship with physical space–time. Computability, despite having exact access to its own discrete data type, provides a unique tool for the investigation of ‘unpredictability’ in both Physics and Biology through its fine-grained analysis of undecidability – note that unpredictability coincides with physical randomness in both classical and quantum frames. Moreover, it now turns out that an understanding of randomness in Physics and Biology is a key component of the intelligibility of Nature. In this paper, we will discuss a few results following along this line of thought.