The results of Cubitt et al. on the spectral gap problem add a new chapter to the issue of undecidability in physics, as they show that it is impossible to decide whether the Hamiltonian of a quantum many-body system is gapped or gapless. This implies, amongst other things, that a reductionist viewpoint would be untenable. In this paper, we examine their proof and a few philosophical implications, in particular ones regarding models and limitative results. In more detail, we examine the (...) way these theorems model many-body quantum systems, and we question what, if anything, is the physical counterpart of the models used by Cubitt et al. We argue that these models are non-representational and that, even if they are so artificial that it is hard to imagine a physical system arising from them, they nonetheless offer an opportunity to learn about the world and the relation between mathematics and reality. On this basis, we draw the conclusion that their results do not undermine the reductionist viewpoint in a strong sense but leave the question open in a weak sense. (shrink)

We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Gödel's first incompleteness theorem within an (axiomatized) theory of gauge fields.

We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.

We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of well-defined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.

The enduring problem of free will has defied resolution across centuries. There is reason to believe that novel factors must be integrated into the analysis to make progress. Within the current physicalist framework, these factors encompass emergence and information theory, in the context of constraints imposed by physical limits on the representation of information. Furthermore the common, but vague, characterization of free will as 'being able to act differently' is rephrased into an explicatum more suitable for formal analysis. It is (...) argued that the mind is an ontologically open system; a causal high-level system, the dynamics of which cannot be reduced to the states of its associated low-level neural systems, not even if it is rendered physically closed. A positive answer to the question of free will for conscious agents is subsequently outlined. (shrink)

The problem of free will has persistently resisted a solution throughout centuries. There is reason to believe that new elements need to be introduced into the analysis in order to make progress. In the present physicalist approach, these elements are emergence and information theory in relation to universal limits set by quantum physics. Furthermore the common, but vague, characterization of free will as "being able to act differently" is, in the spirit of Carnap, rephrased into an explicatum more suitable for (...) formal analysis. It is argued that the mind is an ontologically open system; a causal high-level system, the future of which cannot be reduced to the states of its associated low-level neural systems, not even if it is rendered physically closed. A positive answer to the question of free will is subsequently outlined. (shrink)